REVIEW ARTICLE

Classical dynamical density functional theory: from fundamentals to

applications

Michael te Vrugta, Hartmut Lowenb and Raphael Wittkowskia*

aInstitut fur Theoretische Physik, Center for Soft Nanoscience, WestfalischeWilhelms-Universitat Munster, D-48149 Munster, Germany;bInstitut fur Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universitat Dusseldorf,D-40225 Dusseldorf, Germany

ABSTRACTClassical dynamical density functional theory (DDFT) is one of the cornerstonesof modern statistical mechanics. It is an extension of the highly successful methodof classical density functional theory (DFT) to nonequilibrium systems. Originallydeveloped for the treatment of simple and complex fluids, DDFT is now appliedin fields as diverse as hydrodynamics, materials science, chemistry, biology, andplasma physics. In this review, we give a broad overview over classical DDFT. Weexplain its theoretical foundations and the ways in which it can be derived. The re-lations between the different forms of deterministic and stochastic DDFT as well asbetween DDFT and related theories, such as quantum-mechanical time-dependentDFT, mode coupling theory, and phase field crystal models, are clarified. Moreover,we discuss the wide spectrum of extensions of DDFT, which covers methods withadditional order parameters (like extended DDFT), exact approaches (like powerfunctional theory), and systems with more complex dynamics (like active matter).Finally, the large variety of applications, ranging from fluid mechanics and poly-mer physics to solidification, pattern formation, biophysics, and electrochemistry, ispresented.

PACS CLASSIFICATION05.20.Jj; 05.70.Ln; 47.10.-g; 66.10.Cb; 68.15.+e; 81.30.Fb; 64.70.Pf; 82.35.Jk;64.60.-i; 05.65.+b; 82.70.Dd; 71.15.Mb; 05.30.-d; 05.90.+m 87.10.Ed;

KEYWORDSdynamical density functional theory, DDFT, colloids, soft matter, simple andcomplex fluids, statistical physics

Contents

1 Introduction 4

2 From static to dynamical density functional theory 62.1 Historical overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Static density functional theory . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Variational principle . . . . . . . . . . . . . . . . . . . . . . . . 92.2.2 Approximations for the free energy functional . . . . . . . . . . 11

2.3 Dynamical density functional theory . . . . . . . . . . . . . . . . . . . 14

*Corresponding author: raphael.wittkowski@uni-muenster.de

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3 Standard DDFT 153.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.1 Deterministic DDFT for simple and colloidal fluids . . . . . . . 163.2.2 Stochastic DDFT for simple and colloidal fluids . . . . . . . . . 163.2.3 DDFT for polymer melts . . . . . . . . . . . . . . . . . . . . . 19

3.3 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3.1 Via phenomenological considerations . . . . . . . . . . . . . . . 233.3.2 Via Langevin formalism . . . . . . . . . . . . . . . . . . . . . . 243.3.3 Via Smoluchowski formalism . . . . . . . . . . . . . . . . . . . 253.3.4 Via projection operator formalism . . . . . . . . . . . . . . . . 26

3.4 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.4.1 Limitations of DDFT . . . . . . . . . . . . . . . . . . . . . . . 303.4.2 Tests of DDFT . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.5 Further aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.5.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.5.2 Deterministic or stochastic? . . . . . . . . . . . . . . . . . . . . 343.5.3 Role of the solvent . . . . . . . . . . . . . . . . . . . . . . . . . 363.5.4 Approach to equilibrium . . . . . . . . . . . . . . . . . . . . . . 37

4 Extensions of standard DDFT 384.1 Nonconstant diffusion coefficient . . . . . . . . . . . . . . . . . . . . . 424.2 Hydrodynamic interactions . . . . . . . . . . . . . . . . . . . . . . . . 424.3 Source and sink terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.4 Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.4.1 Potential flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.4.2 Shear flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.5 Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.6 Orientational degrees of freedom . . . . . . . . . . . . . . . . . . . . . 48

4.6.1 Nonspherical particles . . . . . . . . . . . . . . . . . . . . . . . 484.6.2 Magnetic particles . . . . . . . . . . . . . . . . . . . . . . . . . 514.6.3 Active particles . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.7 Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.7.1 Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.7.2 Momentum density . . . . . . . . . . . . . . . . . . . . . . . . . 564.7.3 Kinetic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.8 Nonisothermal systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.8.1 Fixed temperature gradients . . . . . . . . . . . . . . . . . . . 574.8.2 Energy density . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.8.3 Entropy density . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.9 Particle-conserving dynamics . . . . . . . . . . . . . . . . . . . . . . . 594.10 Extensions of DDFT for polymer melts . . . . . . . . . . . . . . . . . . 614.11 Quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5 Exact approaches generalizing DDFT 635.1 General aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.2 Power functional theory . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.2.1 Standard power functional theory . . . . . . . . . . . . . . . . . 645.2.2 Nonspherical and active particles . . . . . . . . . . . . . . . . . 675.2.3 Newtonian mechanics . . . . . . . . . . . . . . . . . . . . . . . 68

2

5.2.4 Quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . 685.3 Projection operator formalism . . . . . . . . . . . . . . . . . . . . . . . 69

5.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.3.2 Projection-operator-based extensions of standard DDFT . . . . 70

5.3.2.1 Extended dynamical density functional theory . . . . 705.3.2.2 Functional thermodynamics . . . . . . . . . . . . . . . 71

5.4 Runge-Gross theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.4.1 Quantum time-dependent density functional theory . . . . . . . 725.4.2 Existence proof for DDFT . . . . . . . . . . . . . . . . . . . . . 74

6 Theories related to DDFT 756.1 Mode coupling theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 756.2 Nonequilibrium thermodynamics . . . . . . . . . . . . . . . . . . . . . 806.3 Phase field crystal models . . . . . . . . . . . . . . . . . . . . . . . . . 816.4 Nonequilibrium self-consistent generalized Langevin equations . . . . . 856.5 Dynamic mean field theory . . . . . . . . . . . . . . . . . . . . . . . . 85

7 Analyzing a DDFT 877.1 Steady solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877.2 Perturbative approaches . . . . . . . . . . . . . . . . . . . . . . . . . . 88

7.2.1 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887.2.1.1 Linear stability analysis . . . . . . . . . . . . . . . . . 887.2.1.2 Front propagation . . . . . . . . . . . . . . . . . . . . 89

7.2.2 Separation of time scales . . . . . . . . . . . . . . . . . . . . . . 897.2.3 Renormalized perturbation theory . . . . . . . . . . . . . . . . 90

7.3 Numerical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

8 Applications 948.1 Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

8.1.1 Colloidal fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . 948.1.2 Atomic and molecular fluids . . . . . . . . . . . . . . . . . . . . 958.1.3 Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 958.1.4 Thin films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 968.1.5 Glassy systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 978.1.6 Granular media . . . . . . . . . . . . . . . . . . . . . . . . . . . 1008.1.7 Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1018.1.8 Driven soft matter . . . . . . . . . . . . . . . . . . . . . . . . . 1018.1.9 Active soft matter . . . . . . . . . . . . . . . . . . . . . . . . . 1028.1.10 Electrochemical systems . . . . . . . . . . . . . . . . . . . . . . 1028.1.11 Biological systems . . . . . . . . . . . . . . . . . . . . . . . . . 104

8.2 Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1058.2.1 Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1058.2.2 Phase separation . . . . . . . . . . . . . . . . . . . . . . . . . . 1068.2.3 Pattern formation . . . . . . . . . . . . . . . . . . . . . . . . . 1078.2.4 Nucleation and solidification . . . . . . . . . . . . . . . . . . . 1088.2.5 Chemical reactions . . . . . . . . . . . . . . . . . . . . . . . . . 1098.2.6 Disease spreading . . . . . . . . . . . . . . . . . . . . . . . . . . 1108.2.7 Feedback control . . . . . . . . . . . . . . . . . . . . . . . . . . 1118.2.8 Sound waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

8.3 Dynamics of the van Hove function . . . . . . . . . . . . . . . . . . . . 112

3

9 Outlook 113

10 Conclusions 114

List of abbreviations

0D zero spatial dimensions1D one spatial dimension2D two spatial dimensions3D three spatial dimensionsABP active Brownian particleBBGKY Bogoliubov-Born-Green-Kirkwood-YvonBD Brownian dynamicsDDFT dynamical density functional theoryDFT density functional theoryDMFT dynamic mean-field theoryEDDFT extended dynamical density functional theoryENE ergodic-to-nonergodicEPD external potential dynamicsFMT fundamental measure theoryFTD functional thermodynamicsGCM Gaussian core modelGDA Generalized diffusion approachMC Monte CarloMCT mode coupling theoryMSR Martin-Siggia-RoseNE-SCGLE nonequilibrium self-consistent generalized Langevin equationsOPD order-preserving dynamicsPCD particle-conserving dynamicsPFC phase field crystalPFT power functional theoryPNP Poisson-Nernst-PlanckRDDFT reaction-diffusion density functional theoryTDDFT time-dependent density functional theoryTR time reversal

1. Introduction

Variational methods have been used in statistical physics for a long time and withtremendous success. Density functional theory (DFT) for quantum systems [1–3], forwhich the Nobel Prize in chemistry 1998 was awarded [4], is one of the central methodsof many-electron theory. It allows for the description of a many-particle system basedon a variational principle in which the one-body density is the only variable. Thesame idea has also been applied to classical fluids [5–7]. Here, a grand-canonical freeenergy functional is minimized by the equilibrium one-body density ρ(~r) with position~r. Classical DFT has found many applications in soft matter physics.

Dynamical extensions of classical DFT, known as “dynamical density functionaltheory” (DDFT) or “time-dependent density functional theory”, were first suggested

4

on a phenomenological basis [8–11] and later derived from the microscopic equationsof motion of the individual particles [12–16]. They describe the time evolution of theone-body density ρ(~r, t) with time t using a continuity equation in which the currentis proportional to the gradient of the functional derivative of the free energy that iszero in the equilibrium case corresponding to static DFT. The resulting equation ofmotion thus describes the relaxation towards the equilibrium state described by DFT.It can, however, also be applied to systems that do not approach equilibrium, suchas driven [17–19] or active [20–23] soft matter. In general, DDFT describes not onlythe equilibrium configuration, but also the dynamical behavior out of equilibrium, inexcellent agreement with Brownian dynamics simulations.

Unlike its static counterpart DFT, DDFT (as presented in Ref. [12]) is not anexact theory. It is based on the adiabatic approximation, which is the assumption thatthe pair correlation of the nonequilibrium system is identical to that of an equilibriumsystem with the same one-body density. While this approximation works well for manysystems, superadiabatic forces, which include memory effects, have been found to beimportant in some cases [24,25]. These can be dealt with using formally exact methodssuch as power functional theory [26–32] or projection operator methods [15,16,33–37].Moreover, the practical application of DDFT requires additional approximations. Thisinvolves, in particular, the choice of the free energy functional, which is not knownexactly and is mostly assumed to have a grand-canonical rather than a canonical form[38,39]. The limitations of DDFT are discussed in Section 3.4.1.

The early forms of DDFT were extended in a vast number of directions, makingthe theory applicable to systems with nonuniform temperature [40], hydrodynamicinteractions [41], or superadiabatic forces [26] and to particles with inertia [42], non-spherical shape [43], or self-propulsion [20]. It has also found a significant number ofapplications, ranging from “typical” ones, such as the derivation of phase field crystal(PFC) models [44], phase separation [14], and solidification [45], to more exotic ones,such as cancer growth [46] or quantum hydrodynamics [47]. DDFT is now used inmany areas of physics, as well as in related subjects such as biology [48–50], chemistry[51–53], materials science [54–56], engineering [57], mathematics [58,59], and philos-ophy [60]. This diversity arises, since the problem of finding a useful and accuratedescription of the collective dynamics in many-particle systems is of importance in alarge number of areas in and beyond physics. In this review, we present this broadrange of applications, together with a detailed discussion of the theoretical foundationsand the various extensions of DDFT.

Our review has the form of a reference work, such that even if it is not read from thebeginning to the end, all important information can be found quickly by consultingthe section one is interested in. Thus, some references are discussed more than once(e.g., a DDFT for active particles with hydrodynamic interactions [22] is relevant forSections 4.2 and 4.6.3), and many sections contain cross-references or longer lists ofarticles. Of course, the article can also be read entirely, which gives a complete overviewover variants and applications of DDFT.

This review is structured as follows: In Section 2, we give a historical overview and abrief introduction to static DFT. We explain the standard forms of DDFT, includingtheir derivation, in Section 3. Extensions of DDFT are presented in Section 4. InSection 5, formally exact approaches that generalize DDFT are discussed. We explaintheories related to DDFT in Section 6. Analytical and numerical methods used toanalyze a DDFT are the topic of Section 7. Applications are reviewed in Section 8.An outlook to possible future work is given in Section 9. We conclude in Section 10.

5

2. From static to dynamical density functional theory

In this section, we first give an overview over the historical development of DDFTin Section 2.1. Afterwards, we provide an introduction to static DFT in Section 2.2,which includes several concepts that are important also for the dynamical case, suchas the free energy functional and the direct correlation function. In Section 2.3, weexplain how DFT can be extended to dynamical situations.

2.1. Historical overview

Pioneering work on the theory of Brownian motion (discovered by Brown [61]) wasdone by Einstein [62, 63], Smoluchowski [64,65], Langevin [66], Fokker [67, 68], andPlanck [69] (although, as pointed out in Ref. [70], some basics were already developedby Lord Rayleigh [71]). The discussion of the inhomogeneous liquid-gas interface by vander Waals [72] is often considered the first density functional study [73]. Further earlycontributions were made by Percus [74], Lebowitz and Percus [75], and Stillinger Jrand Buff [76].

Important foundational work on the statistical mechanics of classical systems, inparticular the theory of ensembles, was done by Gibbs [77]. DFT, however, has itsroots mainly in quantum mechanics. Early functional treatments of electron systemswhere developed by Thomas [78] and Fermi [79]. Modern quantum DFT is based on thefamous results by Hohenberg, Kohn, and Sham [1,2], which allow to describe many-electron systems using a variational principle. An extension to systems at nonzerotemperature was derived by Mermin [3]. Moreover, DFT is applied to atomic nuclei(see Ref. [80] for a short overview over such approaches). Runge and Gross [81] obtaineda dynamical extension of quantum DFT, known as time-dependent density functionaltheory (TDDFT).

The results from Hohenberg and Kohn were applied to classical fluids by Ebner,Saam, and Stroud [5,6]. A transformation between density and one-body potential was,for classical systems, proposed by Yang et al. [7] (influenced by work from De Dominicisand Martin [82]). Another early classical DFT is the generalized van der Waals theory[83,84], further contributions were made by Haymet and Oxtoby [85,86]. Brief historicaloverviews over DFT can be found in Refs. [44,70,73,80,87,88].

Dynamical theories in statistical mechanics that provide a connection between mi-croscopic particle dynamics and macroscopic physics date back to the famous Boltz-mann equation [89] (see Ref. [90] for a historical overview). The Cahn-Hilliard equation[91,92] is an early equation of the gradient dynamics type (see Section 6.2). An exten-sion of the DFT principle towards dynamic equations has been suggested by Evans [8]for spinodal decomposition. This work introduced what is now known as the governingequation of deterministic DDFT (Eq. (35)). Methods of this type were also appliedto spinodal decomposition in Refs. [93,94]. Kirkpatrick and Wolynes [95] employed adynamic DFT in the context of the glass transition (see also Refs. [96,97]). Munakata[11,98–102] derived a dynamical extension of DFT as the overdamped limit of thisresult for application to supercooled liquids (see Sections 3.2.2 and 3.3.1), therebyextending earlier work on this topic [103]. Dieterich et al. [9] presented a phenomeno-logical derivation of DDFT. Nonlinear diffusion equations for freezing were justifiedby the argument that they contain DFT as a static limit [104]. A stochastic dynamicequation for the one-body density of a system of Brownian particles in the form ofDDFT was derived by Dean [105]. Reinel and Dieterich [106] developed a general-

6

ization of classical DFT to time-dependent situations for lattice gases referred to as“time-dependent density functional theory”.

Early approaches to DDFT had in mind a variety of specific applications. They in-cluded solvation dynamics and dipolar molecules. A Smoluchowski-Vlasov equation fororientable particles was obtained by Calef and Wolynes [107], whose results were usedin the study of orientational relaxation by Chandra and Bagchi [108–112]. These meth-ods constituted a first DDFT with orientational degrees of freedom (see Section 4.6.1).Moreover, DDFT had a very close relation to mode coupling theory (MCT) at itsearly stages. Kirkpatrick and Wolynes [95] discussed the relation of a phenomenolog-ical DDFT to MCT. A description of glassy dynamics based on a density functionalHamiltonian was developed by Kirkpatrick and Thirumalai [113]. Kawasaki [114, 115]developed a stochastic DDFT in the form of a Fokker-Planck equation for the proba-bility distribution functional P [ρ], in particular for glassy systems as an alternative tostandard MCT. This method was also formulated in path integral form [116]. Chandraand Bagchi combined DDFT and MCT in theoretical studies of electrolytes [117–120](see Section 8.1.10). An overview over orientational relaxation dynamics is given inRef. [121] and Ref. [122] contains historical remarks. Yoshimori [123] has reviewedearly forms of DDFT. Another important line of development concerns polymers. A“dynamic density functional theory” for polymer dynamics was proposed by Fraaijeand coworkers [10,124] (see also Refs. [125–128]). Kawasaki and Koga [129] applied aDDFT-type model to polymer blends and binary fluids.

Deterministic DDFT was first microscopically derived by Marini Bettolo Marconiand Tarazona [12, 13] from Langevin equations. The theory was later rederived froma Smoluchowski equation by Archer and Evans [14] and from the Mori-Zwanzig pro-jection operator method by Yoshimori [15] and Espanol and Lowen [16].

Standard DDFT has since then been developed in various directions. An existenceproof was given by Chan and Finken [130]. Archer [131] derived a DDFT for atomicfluids, being together with Marini Bettolo Marconi and Tarazona [42] among the firstones to incorporate inertia. Rauscher et al. [132] generalized DDFT to flowing solvents.Hydrodynamic interactions, not present in the original formalism, where included byRoyall et al. [133] and by Rex and Lowen [41]. Further important developments are thetreatment of nonspherical particles [21,43], active particles [20], nonisothermal systems[40,134], and superadiabatic forces [26].

7

1960

1965

1970

1975

1980

1985

1990

1995

2000

2005

2010

2015

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0

5

10

15

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[16]

DD

FT

wit

hsh

ear

flow

by

Bra

der

and

Kru

ger

[137];

DD

FT

for

bia

xia

lpart

icle

sby

Wit

tkow

ski

and

Low

en

[21]

DD

FT

wit

henerg

ydensi

tyby

Wit

tkow

skietal.

[40]

Pow

er

functi

onal

theory

by

Schm

idt

and

Bra

der

[26];

Functi

onal

therm

odynam

ics

by

Anero

etal.

[138]

Part

icle

-conse

rvin

gdynam

ics

by

de

las

Hera

setal.

[38]

Year

Number

ofpublication

sper

year

Figure 1. Important steps in the development of DDFT and publications considered in this review over thecourse of time.

8

A closely related approach are phase field crystal models [139] (see Section 6.3), pro-posed by Elder et al. [135] and later derived from DDFT by van Teeffelen et al. [44].

An overview over the historical development of DDFT is given in Fig. 1 on thepreceding page.

2.2. Static density functional theory

2.2.1. Variational principle

Classical static density functional theory (DFT) is a statistical-mechanical theory forthe equilibrium state of a classical many-body system. We present it following Ref.[140]. Microscopically, a system of N particles is described by the phase-space distribu-

tion function Ψ(~ri, ~pi) that depends on the positions ~ri and momenta ~pi of the

individual particles and is normalized as Tr(Ψ) = 1 with the trace Tr. We introduce a

grand-canonical free energy functional Ω(T, µ, [Ψ]) that depends on the temperatureT and the chemical potential µ, which are treated as fixed parameters, and that ismoreover a functional of the distribution Ψ. In the grand-canonical case, the trace ofan arbitrary function Y on phase space is (for systems in 3D) given by

Tr(Y ) =

∞∑N=0

1

N !h3N

∫d3r1

∫d3p1 · · ·

∫d3rN

∫d3pN Y, (1)

where h is the Planck constant. Denoting the N -particle Hamiltonian by HN , thegrand-canonical free energy functional is given by

Ω(T, µ, [Ψ]) = Tr(Ψ(HN − µN + kBT ln(Ψ))), (2)

where kB is the Boltzmann constant. When evaluated at the equilibrium distribution

Ψeq =1

Ξe−β(HN−µN), (3)

with the thermodynamic beta β = 1/(kBT ) and the grand-canonical partition function

Ξ = Tr(exp(−β(HN − µN))), (4)

the functional (2) is minimized and equal to the actual equilibrium grand-canonical

free energy. If we fix the particle interactions, Ψeq is completely determined by theexternal potential U1(~r).

Since it is defined on a 6N -dimensional phase space, the function Ψ can be verycomplicated. DFT makes it possible to work with the one-body density

ρ(~r) =

⟨ N∑i=1

δ(~r − ~ri)⟩

(5)

instead, where 〈·〉 denotes an ensemble average and δ(~r) is the Dirac delta distribution.The one-body density gives the probability of finding a particle at position ~r. Wedenote the one-body distribution at equilibrium by ρeq(~r). Mermin [3] showed that

9

the external potential U1(~r) entering the equation (3) for Ψeq is uniquely determined

once ρeq(~r) is known, such that Ψeq is a functional of ρeq. Moreover, the mappingρ → U1 can be shown to exist under very general conditions [141]. Hence, we canintroduce a well-defined functional (“intrinsic free energy”)

Fin(T, [ρ]) = Tr(Ψeq[ρ](Hkin +Hint + kBT ln(Ψeq[ρ]))), (6)

with the kinetic part Hkin and the interaction part Hint of the Hamiltonian, that isindependent of the external potential. A further functional is given by

Ω(T, µ, [ρ]) = Fin(T, [ρ]) +

∫d3r ρ(~r)U1(~r)− µ

∫d3r ρ(~r). (7)

This functional is minimal and equal to the equilibrium grand-canonical free energyif evaluated at the equilibrium density ρeq. These considerations lead to the centralvariational principle of DFT

δΩ(T, µ, [ρ])

δρ(~r)

∣∣∣∣ρ=ρeq

= 0. (8)

Ω(T, µ, [ρeq]) then gives the actual grand-canonical free energy of the system [73].For finding the equilibrium configuration, one needs to determine the functional

Ω(T, µ, [ρ]), which in most cases cannot be done exactly. Through a Legendre trans-formation it can be related to a Helmholtz free energy functional F (T, [ρ]) in the form[139]

Ω(T, µ, [ρ]) = F (T, [ρ])− µ∫

d3r ρ(~r). (9)

We can split the free energy into three parts:

F (T, [ρ]) = Fid(T, [ρ]) + Fexc(T, [ρ]) + Fext([ρ]). (10)

The first term

Fid(T, [ρ]) = kBT

∫d3r ρ(~r)(ln(Λ3ρ(~r))− 1), (11)

with the thermal de Broglie wavelength Λ, is the exact expression for the free energyof an ideal gas. An external potential U1(~r) is taken into account through the last term

Fext([ρ]) =

∫d3r ρ(~r)U1(~r). (12)

Finally, the term Fexc(T, [ρ]), known as excess free energy, gives the contribution fromthe particle interactions. In general, it cannot be calculated exactly and needs tobe approximated. Approximation methods (which are also required in DDFT) arediscussed in Section 2.2.2. Note that many authors refer to the intrinsic free energyfunctional Fin given by Eq. (6) as “free energy” and use the symbol F for the intrinsicfree energy. This is simply a different convention. It is, however, important to clarify

10

whether F denotes (in our terminology) the intrinsic or the full free energy, since thisaffects the form of the DDFT equation (see Section 2.3). In this review, we employthe convention (10). From now on, we will drop the parametric dependence on T inour notation.

Reviews of DFT (some of which include comments on DDFT) can be found in Refs.[140,142–148]. Books discussing DFT include Refs. [149,150].

2.2.2. Approximations for the free energy functional

For practical applications of DFT, one requires an expression for the free energy [151].More precisely, since the ideal gas contribution Fid and the external potential con-tribution Fext are known analytically, one has to find a good approximation for theexcess free energy Fexc. Since the same problem arises in DDFT, which is in mostcases based on a free energy functional from DFT, we here explain various ways ofconstructing an approximate excess free energy.

A useful starting point are functional Taylor expansions of the free energy arounda homogeneous reference density ρ0, which we present following Ref. [139]. They takethe form

Fexc[ρ] = F (0)exc(ρ0)−

∞∑n=1

kBT

n!

∫d3r1 · · ·

∫d3rn c

(n)(~r1, . . . , ~rn)

n∏i=1

∆ρ(~ri) (13)

with an irrelevant constant F(0)exc(ρ0), the n-th-order direct correlation function

c(n)(~r1, . . . , ~rn) = − 1

kBT

δnF [ρ]

δρ(~r1) · · · δρ(~rn)

∣∣∣∣ρ=ρ0

, (14)

and the reduced density ∆ρ(~r) = ρ(~r)− ρ0. Truncating the expansion (13) at secondorder gives the Ramakrishnan-Yussouff approximation [152]

Fexc[ρ] = −1

2kBT

∫d3r1

∫d3r2 c

(2)(~r1 − ~r2)∆ρ(~r1)∆ρ(~r2), (15)

which allows to predict the freezing transition. The correlation function can be ob-tained, e.g., from the random phase approximation

c(2)(~r1 − ~r2) = −U2(~r1 − ~r2)

kBT(16)

or the virial expression

c(2)(~r1 − ~r2) = exp

(− U2(~r1 − ~r2)

kBT

)− 1. (17)

Which approximation gives good results depends on the system under consideration.A very simple form of Fexc that is particularly useful in the case of soft interactionsand high densities is the mean-field approximation [149,153–157]

Fexc =1

2

∫d3r

∫d3r′ U2(~r − ~r′)ρ(~r)ρ(~r′). (18)

11

Since the second functional derivative of Fexc gives the direct pair-correlation function,the mean-field approximation generates the random phase approximation (16). On theother hand, the form (18) can be obtained by inserting the random phase approxi-mation (16) into the Ramakrishnan-Yussouff approximation (15). (This also requiresthe replacement ∆ρ → ρ, which is possible in DDFT on this level of approximation,since the difference arising from this replacement vanishes in the DDFT equation (35)due to a combination of functional derivatives and gradient operator.) The mean-fieldapproximation has been used, e.g., in DDFT models for ultrasoft particles [17], activeparticles [23], and social interactions [50].

For particles with hard-core interactions, such as hard spheres, fundamental measuretheory (FMT), which has been introduced by Rosenfeld [158], is a very successfulapproach. We present it here following Ref. [159]. A review can be found in Ref. [160].Note that FMT can also be combined with the mean-field approximation (18): Forexample, when describing particles interacting through a combination of hard-sphererepulsion and attractive interaction, one can approximate Fexc as a sum of a termfor the hard-sphere interaction (obtained from FMT) and a mean-field term for theattractive interaction [14].

The starting point is the fact that the excess free energy for a dilute hard-spheremixture (with Ns species) can be written as

Fexc = kBT

∫d3r (n0(~r)n3(~r) + n1(~r)n2(~r)− ~n1(~r) · ~n2(~r)) (19)

with the weighted densities

nα(~r) =

Ns∑i=1

∫d3r′ ρi(~r′)wiα(~r − ~r′) (20)

and the weight functions

wi3(~r) = Θ(Ri − ‖~r‖), (21)

wi2(~r) = δ(Ri − ‖~r‖), (22)

wi1(~r) =wi2

4πRi, (23)

wi0(~r) =wi2

4πR2i

, (24)

~wi2(~r) =~r

rwi2, (25)

~wi1(~r) =~wi2

4πRi, (26)

where Θ(·) denotes the Heaviside step function, Ri is the radius of the spheres ofthe i-th species, and ‖·‖ is the Euclidean norm. The form (19) is extended to largerdensities by the ansatz

Fexc = kBT

∫d3r gfmt(nα(~r)) (27)

12

with the function gfmt that has to recover the low-density limit. Various forms of gfmt

are used in the literature. Rosenfeld [158] proposed the form

gfmt = −n0 ln(1− n3) +n1n2 − ~n1 · ~n2

1− n3+n3

2 − 3n2~n2 · ~n2

24π(1− n3)2. (28)

Numerical aspects of this expression are discussed in Ref. [161]. This form, however,cannot describe hard-sphere crystals [159]. A problem occurs due to a divergence inthe last term on the right-hand side of Eq. (28). This can be avoided using the q3

correction [162,163], in which the last term of Eq. (28) is modified as

1

24π(1− n3)2

(n2 −

~n2 · ~n2

n2

)3

. (29)

An alternative proposed by Tarazona [164] is to introduce the tensorial weight function(here in the notation of Refs. [160,165])

wim2(~r) =

(~r ⊗ ~rr2− 1

31

)wi2(~r) (30)

with the dyadic product ⊗ and the 3D identity matrix 1. From wim2one obtains a

tensorial weighted density nm2, which allows to rewrite the third term on the right-

hand side of Eq. (28) as [160]

n32 − 3n2~n2 · ~n2

24π(1− n3)2+

9(~n2 · nm2~n2 − 1

2 Tr(n3m2

))

24π(1− n3)2, (31)

where Tr denotes the trace of a matrix (and not the trace on phase space as in the restof this article). Using the q3 correction or the tensorial modification, a description ofhard-sphere crystals in FMT is possible [159]. Additional modifications of FMT havealso been proposed, such as the White Bear FMT [166]. A state-of-the-art form isWhite Bear mark II [167], given by [159]

gfmt = −n0 ln(1− n3) +n1n2 − ~n1 · ~n2

1− n3

(1 +

1

3g2(n3)

)+n3

2 − 3n2~n2 · ~n2

24π(1− n3)2

(1− 1

3g3(n3)

) (32)

with the functionals

g2(n3) =1

n3(2n3 − n2

3 + 2(1− n3) ln(1− n3)), (33)

g3(n3) =1

n23

(2n3 − 3n23 + 2n3

3 + 2(1− n3)2 ln(1− n3)). (34)

When combined with a tensorial modification, White Bear mark II gives accurate re-sults in the description of hard-sphere crystals [168]. FMT can be extended to generalnonspherical convex particle shapes [169–174], which allows to apply it to orienta-tional dynamics [175,176]. An FMT-based DDFT for hard polyhedra was developed

13

by Marechal and Lowen [177]. Moreover, lattice FMT [178–180] can be used in latticeDDFT models of hard rods [181]. Applications of FMT to DDFT can be found, e.g.,in Refs. [23,38,41,53,133,159,175,176,182–210].

2.3. Dynamical density functional theory

Up to now, we have only considered equilibrium situations, in which the density ρ(~r)is constant with respect to time and minimizes the grand-canonical free energy. In thiscase, the density distribution can be determined from the variational principle (8).The central idea of DDFT is to extend this principle to nonequilibrium situations. Ifthe equilibrium state is the one in which the free energy is minimized, the dynamicequation will have a form which ensures that the free energy decreases over time.This allows to relate the rate of change ∂tρ(~r, t) of the particle number density to thevariation of the free energy. Moreover, since ρ(~r, t) is a conserved quantity, its timeevolution will have the form of a continuity equation, i.e., ∂tρ(~r, t) is proportional tothe gradient of a flux. These considerations lead to the central equation of deterministicdynamical density functional theory

∂

∂tρ(~r, t) = Γ~∇ ·

(ρ(~r, t)~∇ δF [ρ]

δρ(~r, t)

)(35)

with the mobility Γ. Note that the theory is formally based on the free energy F , incontrast to DFT which uses the grand-canonical free energy Ω. However, this makes nosignificant difference in most practical applications, and one typically uses free energyfunctionals from DFT, which are grand-canonical (see Section 4.9 for exceptions). If Fdenotes the intrinsic free energy (see Section 2.2 for a discussion of this convention), a

term Γ~∇·(ρ~∇U1) has to be added to the right-hand side of Eq. (35). The general result(35) can be derived in a variety of ways (see Section 3.3 for an overview). Some authorsadd a noise term to Eq. (35), this corresponds to the stochastic DDFT equation

∂

∂tρ(~r, t) = Γ~∇ ·

(ρ(~r, t)~∇ δF [ρ]

δρ(~r, t)

)+ ~∇ · (

√2ΓkBTρ(~r, t)~η(~r, t)), (36)

where ~η(~r, t) is a multiplicative noise. Deterministic DDFT describes the ensemble-averaged one-body density, whereas stochastic DDFT is concerned either with theexact microscopic or with a coarse-grained density [211]. The relation of deterministicand stochastic approaches is discussed in Section 3.5.2.

At the heart of deterministic DDFT is the assumption that the equal-time two-pointcorrelation function of the nonequilibrium system is identical to that of the equilibriumsystem with the same one-body density [12]. This adiabatic approximation, which isstrictly valid only if the system density evolves infinitely slowly (quasi-statically), al-lows to express the interaction term in the dynamic equation in terms of the functionalderivative of the excess free energy. For obtaining the time evolution of ρ(~r, t) fromEq. (35), one needs to determine the form of F [ρ], which, in general, cannot be doneexactly. In many situations, one uses the equilibrium density functional, making theapproximation that it can still be applied close to equilibrium. This approximationcan break down in the case of, e.g., driven or active systems, which require the con-struction of a different functional. Note that the adiabatic approximation made in thederivation of stochastic DDFT from underdamped dynamics (see Section 3.3.1) is the

14

weaker assumption that the density is a slow variable compared to the momentum.In this review, we denote by adiabatic approximation the assumption of equilibriumpair correlations unless stated otherwise. Stochastic DDFTs describe a coarse-grainedrather than an ensemble-averaged density, and can thus differ in their degree of coarsegraining. If they are set up as an exact (as compared to Brownian dynamics) theoryfor the microscopic density operator, they have no adiabatic approximation. Similarapproximations can, however, be involved in further coarse graining [211].

The dynamical principle of DDFT can be expressed in four ways:

(1) As a (deterministic or stochastic) differential equation for the one-body densityρ(~r, t) in the form (35) or (36).

(2) As a variational principle based on a dissipation functional (see Section 6.2).(3) As a stochastic Fokker-Planck equation for the probability distribution P [ρ] (see

Section 3.2.2).(4) As an action functional in the path integral formalism (see Section 7.2.3).

The formulation as a differential equation is by far the most common one, althoughothers can have advantages in specific contexts (such as the dissipation functionalfor calculating entropy production [212] or the action functional for renormalization[213]). The last two ways are associated with stochastic DDFT.

Introductions to DDFT can be found in Refs. [143,153,214,215].

3. Standard DDFT

3.1. Overview

In the standard form of DDFT, the equation of motion takes the form (35) for thedeterministic and (36) for the stochastic case. This equation describes the dynamicsof the one-body density ρ(~r, t) of a fluid. Standard DDFT assumes that the particlesinteract through a two-body potential U2(~r1, ~r2) = U2(~r1 − ~r2) and have no degreesof freedom other than their center-of-mass position. (N -body interactions can alsobe incorporated [14], this is discussed in Section 3.3.3.) In practice, the pair poten-tial will generally be assumed to have the form U2(‖~r1 − ~r2‖), such that standardDDFT describes spherical particles. (Strictly speaking, this assumption is made forthe interaction potential and not for the physical particles. For example, the effectiveinteraction between polymers is often assumed to have a Gaussian form [155,157]. Thesame form has been used for a DDFT model of humans [50]. A Gaussian interactionpotential has spherical symmetry, but this does not mean that polymers or humansare spheres.) Hydrodynamic interactions and effects of inertia are neglected and thesystem is assumed to be isothermal. In this section, we will cover deterministic (seeSection 3.2.1) and stochastic (see Section 3.2.2) DDFT for simple and colloidal fluidsand DDFT for polymer melts (see Section 3.2.3) as these are the three main concep-tually distinct lines of development. The deterministic form (35) is the most widelyused one, such that we will, for simplicity, often write “(standard) DDFT” to referonly to this theory if there is no danger of confusion.

Equation (35) describes, for time-independent external fields, a diffusive relaxationto an equilibrium state. In the classification by Hohenberg and Halperin [216], it is amodel B equation, i.e., it describes a conserved order parameter in the overdampedlimit without hydrodynamics [217]. The fact that Eqs. (35) and (36) are overdampedcan have two origins. First, it is possible that the underlying dynamics is itself over-

15

damped, which is the case if it is given by overdamped Langevin equations (as in thederivations by Marini Bettolo Marconi and Tarazona [12] and Dean [105]). In thiscase, the underlying particles do (within the theory) not have momentum degrees offreedom, such that a momentum density cannot even be defined. Second, it is possiblethat the underlying dynamics is underdamped (e.g., Hamiltonian) such that a fulltheory would also contain the momentum density. In this case, one makes the furtherapproximation that the momentum density relaxes quickly compared to the numberdensity on the level of the mesoscopic theory (as done by Munakata [11]).

The main difference of Eq. (35) to the standard diffusion equation

∂

∂tρ(~r, t) = D~∇2ρ(~r, t) (37)

with the diffusion constant D = ΓkBT , to which Eq. (35) reduces for noninteractingparticles (ideal gas free energy), is that interactions of the particles with each otherand with external fields can also be incorporated [218].

3.2. Variants

3.2.1. Deterministic DDFT for simple and colloidal fluids

The deterministic form of DDFT was derived microscopically for colloidal fluids byMarini Bettolo Marconi and Tarazona [12] as well as Archer and Evans [14] and forsimple fluids by Archer [131]. These theories and their derivations are discussed indetail in Section 3.3.2, Section 3.3.3, and Section 4.7.1, respectively.

3.2.2. Stochastic DDFT for simple and colloidal fluids

Stochastic DDFT was developed by Kawasaki [114, 115, 219], Munakata [11, 98, 101],Kirkpatrick and Wolynes [95], and Dean [105], and is thus historically older thandeterministic DDFT. The diversity of its roots can make this form difficult to access,since there are approaches that are equivalent even though they look very different(such as the Fokker-Planck and the path-integral formalism by Kawasaki), but alsoapproaches that are not equivalent despite looking identical (such as the stochasticDDFT equations by Kawasaki and Dean).

A pioneer of stochastic DDFT was Munakata [11], whose derivation is discussed inSection 3.3.1. This approach, which starts from an equation presented by Kirkpatrickand Wolynes [95] and from which earlier theories for supercooled liquids [103] canbe recovered, allows to study density fluctuations in liquids [98,99,220,221], polymerconformation [222,223], and shear viscosity [224,225]. Within this framework, variousH-theorems can be proven [101].

Kawasaki’s DDFT, first derived in Ref. [114], is a stochastic theory for the coarse-grained probability distribution functional P ([ρ], t) and applicable to both simple andcomplex fluids. This method has been developed as an alternative to and extension ofMCT for the description of supercooled liquids and the glass transition [226–231]. Itcan be solved by mapping it onto a kinetic lattice gas model [226,227,232]. A furtherdiscussion can be found in Ref. [233]. We present the derivation following Ref. [234]:In the case of a colloidal fluid, one starts from the Smoluchowski equation

∂

∂tΨ(~ri, t) = LS(~ri)Ψ(~ri, t) (38)

16

for the many-particle distribution function Ψ with the Smoluchowski operator

LS(~ri) =

N∑i=1

D~∇~ri ·(~∇~ri +

1

kBT~∇~riUN (~ri)

)(39)

with the total potential energy UN (~ri).The system is then coarse-grained by splitting it into cells of volume vρ. To over-

come the problem that the interaction UN depends on distances smaller than thecoarse-graining size, one first makes vρ infinitesimal, which allows to rewrite the in-teraction term in Eq. (38) using a bilinear functional U(ρ). It is then assumed that,after the coarse graining, one can express U as a bilinear in the coarse-grained den-sity ρ. Expanding around the homogeneous liquid state ρ = ρ0 up to second order in∆ρ = ρ(~r) − ρ0 and using that Fourier transforming and averaging the second-orderterm gives the static structure factor and therefore the direct pair-correlation functionc(2)(r), one finds [234]

∂

∂tP ([ρ], t) = LFP[ρ]P ([ρ], t) (40)

with the Fokker-Planck operator

LFP[ρ] = − D

kBT

∫d3r

δ

δρ(~r, t)~∇ ·(ρ(~r, t)~∇

(kBT

δ

δρ(~r, t)+

δF [ρ]

δρ(~r, t)

))(41)

and the coarse-grained free energy

F [ρ] = kBT

∫d3r ρ(~r, t) ln

(ρ(~r, t)

ρ0− 1

)− 1

2kBT

∫d3r

∫d3r′ c(2)(‖~r − ~r′‖)∆ρ(~r, t)∆ρ(~r′, t).

(42)

Note that Eq. (40) has a stationary solution of the form exp(−βF [ρ]) [11]. Equation(40) is the Fokker-Planck equation corresponding to the stochastic dynamic equation

∂

∂tρ(~r, t) = Γ~∇ ·

(ρ(~r, t)~∇ δF [ρ]

δρ(~r, t)

)+ ~∇ · (

√2ΓkBTρ(~r, t)~η(~r, t)) (43)

with the Gaussian noise ~η(~r, t) satisfying

〈ηi(~r, t)〉 = 0, (44)

〈ηi(~r, t)ηj(~r′, t)〉 = δijδ(~r − ~r′)δ(t− t′). (45)

Since Eq. (43) is simply the Langevin equation corresponding to the Fokker-Planckequation (40) [211], the density field ρ(~r, t) has the same interpretation (spatiallycoarse-grained density profile) in both equations. One can, of course, make the coarse-graining cells very small, such that they are comparable to the interparticle distance, inwhich case the density field would be a very “spiky” and locally smoothened functionfor which noise is very important [114].

17

A coarse-grained description can also be derived for an atomic fluid. Here, themicroscopic description is given by the Liouville rather than by the Smoluchowskiequation. As a starting point, the method of fluctuating hydrodynamics is used, whichprovides a coarse-grained Fokker-Planck equation for a functional P ([ρ,~g], t) that alsodepends on the momentum density ~g(~r, t). Using the projection operator method [34],the momentum density is then eliminated, giving a closed equation of motion forP ([ρ], t) that, despite describing a different physical system, also has the form (40).The only difference is that D/(kBT ) is replaced by τ/m, where τ is the time scaleof momentum relaxation that determines the time scale on which the coarse-graineddescription is valid [114,234] and m is the particle mass. A detailed discussion of thederivation of stochastic DDFT can also be found in Ref. [115]. Reviews are given byRefs. [234,235].

A closely related approach is the (generalized) stochastic Smoluchowski equation[70,236] (see also Ref. [237])

∂

∂tρ(~r, t) = ~∇ ·

(Γg(ρ)~∇ δF [ρ]

δρ(~r, t)

)+ ~∇ · (

√2ΓkBTg(ρ)~η(~r, t)) (46)

with the function g(ρ). This approach (which also exists in a deterministic form) isbased on the idea of modeling the dynamics of the density using, instead of Γρ andD, the more general expressions Γg(ρ) and Dg(ρ) with functions g(ρ) and g(ρ) thattake into account microscopic constraints (the standard case is recovered by settingg(ρ) = ρ and g(ρ) = 1). For the same purpose, the standard Boltzmann entropyis replaced by a generalized entropy, which also leads to a modified free energy. Ageneralized Einstein relation, derived from the requirement that an equilibrium stateof the generalized Smoluchowski equation is a thermodynamic equilibrium state asdescribed by the generalized free energy, shows that the second derivative of the localdensity of the generalized entropy is proportional to the ratio g(ρ)/g(ρ). This allowsto eliminate g(ρ) and to write the dynamics in the form (46). Whether a noise term isrequired depends on whether ρ describes the coarse-grained or the ensemble-averageddensity (see Section 3.5.2) [70].

Another form of stochastic DDFT was developed by Dean [105]. It was derived fora system of N overdamped particles in a white-noise heat bath interacting via a pairpotential U2(~r). Using stochastic calculus, he showed that the density operator

ρ(~r, t) =

N∑i=1

δ(~ri(t)− ~r) (47)

with the positions ~ri of the individual particles obeys the Langevin equation

∂

∂tρ(~r, t) = ~∇ · (

√2ΓkBT ρ(~r, t)~η(~r, t))

+ Γ~∇ ·(ρ(~r, t)

∫d3r′ ρ(~r′, t)~∇U2(~r − ~r′)

)+D~∇2ρ(~r, t).

(48)

18

With the free energy

F [ρ] =1

2

∫d3r

∫d3r′ ρ(~r, t)U2(~r−~r′)ρ(~r′, t)+kBT

∫d3r ρ(~r, t)(ln(Λ3ρ(~r, t))−1), (49)

Eq. (48) becomes

∂

∂tρ(~r, t) = Γ~∇ ·

(ρ(~r, t)~∇ δF [ρ]

δρ(~r, t)

)+ ~∇ · (

√2ΓkBT ρ(~r, t)~η(~r, t)), (50)

making the relation to DDFT more obvious. (In Dean’s original article [105], the lastterm of Eq. (49) is simply written with ρ ln(ρ) instead of ρ(ln(Λ3ρ)−1), which makes nodifference for Eq. (50).) Mathematically, the notation in Eqs. (48)-(50), which involvessquare roots and logarithms of Dirac delta distributions, should be understood asformal [238]. An extension to orientational degrees of freedom is also possible [239]. Anotable aspect in the derivation of Eq. (50) is the treatment of the noise: The Langevinequations (72) (see below), which are the starting point for the derivation of Eq. (50),contain an additive noise term ~χi(t) for each particle i. This leads to a noise termin the dynamic equation for ρ in which all ~χi(t) appear. Then, this term is rewrittenin such a way that it contains the multiplicative noise field ~η(~r, t) instead, giving theclosed equation of motion (50).

Dean’s theory is a formally exact microscopic balance equation for the density fieldρ(~r, t) defined as a sum over Dirac delta distributions, in contrast to both deter-ministic DDFT and to Eq. (43), which provide coarse-grained dynamic equations. Inconsequence, Eqs. (43) and (50) should, despite the fact that they look identical, notbe confused [233,240]. Equation (50) can also be written as a Fokker-Planck equation,which takes the form of Eq. (40) (although, again, the meaning is different). Often,the name Dean-Kawasaki equation is used for both Eq. (50) and Eq. (43) (although,given that they are not equivalent, this name is not ideal [233]). The Dean-Kawasakiequation (in one or both of its forms) is discussed in more detail in Refs. [241–247].Moreover, Dean’s results are of importance in active matter physics [248–255].

For mathematical discussions of the well-posedness of stochastic differential equa-tions of this type, see Refs. [58,59]. The stress tensor of stochastic DDFT is discussed inRef. [256]. Extensions to underdamped dynamics that include the momentum densityare possible [257–259]. A third way of setting up stochastic DDFT in addition to usinga Langevin or a Fokker-Planck equation is the path integral formalism [116,260–262](see Section 7.2.3 for details).

3.2.3. DDFT for polymer melts

DDFT also plays an important role in the study of polymers. The theory was first pre-sented by Fraaije [10] (see also Ref. [125]) and then improved significantly by Fraaijeet al. [124]. Earlier work on models of this type was done by Binder [263] and Kawasakiand Sekimoto [264, 265, 266]. Although there are similarities to the DDFTs presentedin Sections 3.2.1 and 3.2.2, differences arise from the fact that polymers are chains ofconnected beads rather than simple particles. This form of DDFT is also known as“dynamic mean-field density functional method” and typically introduced as gener-alized time-dependent Ginzburg-Landau theory [124]. Historically, the methods fromSections 3.2.1 and 3.2.2 and the method presented in this section have had a relativelyindependent development.

19

Consider a mixture of N chains of length nc, which consist of Npt particle typesindexed by an integer i. For each particle type, one introduces a density field ρi(~r, t).The general evolution equation, obtained by Kawasaki and Sekimoto [264], reads

∂

∂tρi(~r, t) = −

Npt∑j=1

∫d3r′Mij(~r, ~r

′, t)µj(~r′, t) + ηi(~r, t) (51)

with the kinetic coefficient matrix Mij , the chemical potential

µi(~r, t) =δF [ρk]δρi(~r, t)

, (52)

and the noise ηi (which we will drop from now on). The microscopic construction ofpolymer DDFT is discussed in Ref. [267]. Various approximations for the collectivedynamics can be used [268]. The simplest one, employed by Fraaije [10,124], is thelocal coupling

∂

∂tρi(~r, t) =

Dlo

kBT~∇ · (ρi(~r, t)~∇µi(~r, t)) (53)

with the diffusion constantDlo. Here, the close relation to the standard DDFT equation(35) is obvious. The introduction of an additional pressure functional allows to accountfor incompressibility [124], which gives for a binary mixture

∂

∂tρa(~r, t) = Mbνm~∇ ·

(ρa(~r, t)ρb(~r, t)~∇(µa(~r, t)− µb(~r, t))

), (54)

∂

∂tρb(~r, t) = Mbνm~∇ ·

(ρa(~r, t)ρb(~r, t)~∇(µb(~r, t)− µa(~r, t))

)(55)

with the bead mobility parameter Mb and the constant molecular volume νm. Morecomplex approaches are the Rouse dynamics

∂

∂tρi(~r, t) =

Dro

kBT

Npt∑j=1

~∇~r ·∫

d3r′ Pij(~r, ~r′, t)~∇~r′µj(~r′, t) (56)

with the diffusion constant Dro and the reptation dynamics for continuous chainsparametrized by a real index s [265]

∂

∂tρs(~r, t) = − Dc

kBT

∫d3r′

∫ nc

0ds′(

∂2

∂s∂s′Pss′(~r, ~r

′, t))µs′(~r

′, t) (57)

with the diffusion constant Dc. The two-body correlators are defined as [268]

Pss′(~r, ~r′, t) = N〈δ(~r − ~rs)δ(~r′ − ~rs′)〉, (58)

Pij(~r, ~r′, t) =

nc∑s=1

nc∑s′=1

δKisδ

Kjs′Pss′(~r, ~r

′, t), (59)

20

where ~rs is the position of bead s and δKij is a Kronecker function that is 1 if bead j

is of type i and 0 otherwise. (This means that δKij is not a standard Kronecker delta.

For example, if we have two particle types which we call 1 and 2, δK11 would be zero

if the first bead is not of type 1.) Note that in the literature, time-dependence issometimes dropped, i.e., one writes, e.g., ∂tρ(~r) instead of ∂tρ(~r, t) even though ρ istime-dependent.

The statistical theory is introduced following Ref. [124]. Microscopically, the system

is described using a distribution function Ψ(~r11, . . . , ~rNnc) with ~rls being the position

of bead s from chain l, i.e., we now also indicate the chain number. An average of themicroscopic density operator is given by

ρi[Ψ](~r) =

N∑l=1

nc∑s=1

δKis Tr(Ψδ(~r − ~rls)), (60)

where the trace denotes an integral over RNnc . For an observed density ρi(~r), the

requirement ρi(~r) = ρi[Ψ](~r) gives an equivalence class of microscopic distributionfunctions on which a free energy functional is defined. It is now assumed that (a)correlations between chains can be accounted for in the free energy by a mean-fieldapproximation (while the chains are treated exactly) and that (b) the distribution

function Ψ is chosen in such a way that the free energy is minimized. Ψ is thusindependent of the history of the system and only constrained by the density. Thisallows to obtain a bijective relation between density fields and external potentials.Denoting the external potential for species i by Ui(~r), the constrained minimizationbased on DFT gives the intrinsic free energy functional

Fin[ρ] = −kBT ln(Zgc) + kBT ln(N !)−Npt∑i=1

∫d3r ρi(~r)Ui(~r) + Fni[ρ] (61)

with the partition function of a single Gaussian chain Zgc and the nonideal free energyFni.

DDFT for polymer melts is reviewed in Refs. [269,270]. Extensions are discussed inSection 4.10 and applications in Section 8.1.3.

3.3. Derivation

In this subsection, we discuss how DDFT can be derived from the microscopic particledynamics or from phenomenological considerations. While we focus on the determin-istic form of DDFT, we also include the result by Munakata [11]. The four main waysare a phenomenological derivation [9,11], a derivation from Langevin equations [12],a derivation from a Smoluchowski equation [14], and a derivation from the projectionoperator formalism [15,16]. They are visualized in Fig. 2 on the next page. Langevinequations and Smoluchowski equation are statistically equivalent descriptions of theoverdamped motion of Brownian particles, from which the dynamics of the one-bodydensity is obtained using ensemble averages and an adiabatic approximation. Thederivation starting from the projection operator formalism, in contrast, bases on thedeterministic Hamiltonian dynamics of the individual particles and introduces the as-sumption of overdamped motion at a later stage. By coarse graining, the Langevin

21

increasing length and time scales

Smoluchowskiequation

Langevin equations

Hamiltonianequations

Dynamicaldensity

functionaltheory

Phenome-nologicallevel

statisticallyequivalent

coarsegraining

projection operator formalism

phenomenologicalconsiderations

Figure 2. Different ways of deriving DDFT.

equations can be derived from the Hamiltonian equations. Note that projection opera-tors have also been used to derive stochastic DDFT (see Section 3.2.2) from stochasticdescriptions in the form of fluctuating hydrodynamics [114] or the Smoluchowski equa-tion [271].

A detailed conceptual understanding of the different types of derivations is of highimportance especially for the development of further extensions of DDFT. These areoften derived along the same lines as standard DDFT. For example, Rex and Lowen [41]obtained a DDFT for particles with hydrodynamic interactions (see Section 4.2) fromthe underlying Smoluchowski equation by generalizing the derivation by Archer andEvans [14] (see Section 3.3.3) and Wittkowski et al. [40] used the projection operatorformalism to derive a DDFT for nonisothermal mixtures (see Paragraph 5.3.2.1) basedon the results by Espanol and Lowen [16] (see Section 3.3.4). When extending anexisting derivation, it is important to understand where in this derivation the essentialapproximations are made, which in DDFT concerns the adiabatic approximation. Thisis a closure relation whose precise effect can vary between different derivations. In thesimplest scenario, a derivation starts from already overdamped Brownian dynamics [12,14] (see Sections 3.3.2 and 3.3.3). The adiabatic approximation then allows to expressthe unknown pair-correlation function in terms of the one-body density by assumingthat the pair-correlation has the form it would have in an equilibrium system with thesame one-body density. Alternatively, one can take as a microscopic starting point asystem of underdamped particles, governed by Hamilton’s equations or underdampedLangevin equations. In this case, the adiabatic approximation can appear in two forms:Some authors, such as Munakata [11] (see Section 3.3.1) or Espanol and Lowen [16](see Section 3.3.4), employ an adiabatic approximation of a slightly different form toeliminate momentum degrees of freedom in order to arrive at an overdamped diffusiveequation of motion. Other authors, such as Archer [272] (see Section 4.7.1), use thestandard adiabatic approximation to evaluate an unknown correlation function also inthis case, which leads to a DDFT equation that involves inertia. Finally, the adiabaticdynamics can depend on whether or not the conservation of particle order is takeninto account [273].

22

3.3.1. Via phenomenological considerations

The fastest way of deriving the DDFT equation is to use phenomenological considera-tions. Among the first to derive DDFT was Munakata [11]. The starting point are thehydrodynamic equations [95]

∂

∂tρ(~r, t) = − 1

m~∇ · ~g(~r, t), (62)

∂

∂t~g(~r, t) = −ρ(~r, t)~∇ δF

δρ(~r, t)−∫

d3r′∫ t

0dt′M(~r, ~r′, t− t′)~g(~r′, t′) +~f(~r, t) (63)

with the particle mass m, momentum density ~g, dissipative matrix M, and randomforce ~f. When one assumes that

M(~r, ~r′, t− t′) = 2M01δ(~r − ~r′)δ(t− t′) (64)

with a constant M0, Eq. (63) simplifies to

∂

∂t~g(~r, t) = −ρ(~r, t)~∇ δF

δρ(~r, t)−M0~g(~r, t) +~f(~r, t). (65)

An adiabatic approximation, here corresponding to the assumption that ρ is the onlyslow variable (such that ~g relaxes quickly and we can set ∂t~g = ~0), gives from Eq. (65)

~g(~r, t) =1

M0

(− ρ(~r, t)~∇ δF

δρ(~r, t)+~f(~r, t)

). (66)

In this case, since we have started from the underdamped equations (62) and (63), theadiabatic approximation is required to obtain a closed diffusive (first order in temporalderivatives) equation of motion for ρ(~r, t). Inserting Eq. (66) into Eq. (62) gives theDDFT equation

∂

∂tρ(~r, t) =

1

mM0

~∇ ·(ρ(~r, t)~∇ δF

δρ(~r, t)+~f(~r, t)

), (67)

which can also be written as a Fokker-Planck equation (see Section 3.2.2).A different type of phenomenological derivation was presented by Evans [8] and later

by Dieterich et al. [9] (similar arguments were employed by Fraaije et al. [124]). Insteadof the hydrodynamic equations (62) and (63), the starting point is the generalizedFick’s law

~J(~r, t) = −Γρ(~r, t)~∇µ(~r, t) (68)

with the current ~J , where the gradient of the chemical potential µ is used as thethermodynamic driving force. As long as µ is not specified, the form (68) does notexclude, e.g., advection by a flow field, since µ may contain a velocity potential (thispossibility is discussed in Section 4.4.1). To obtain DDFT, we need to assume thatthe function µ is a chemical potential that is constructed on the basis of equilibrium

23

theory. In static DFT, µ is given by

µ =δF [ρ]

δρ(~r). (69)

Together with the continuity equation for conserved ρ

∂

∂tρ(~r, t) + ~∇ · ~J(~r, t) = 0, (70)

we obtain the DDFT equation

∂

∂tρ(~r, t) = Γ~∇ ·

(ρ(~r, t)~∇ δF [ρ]

δρ(~r, t)

). (71)

This is a nonlinear diffusion equation that reduces to the ordinary diffusion equation(37) in the ideal-gas limit.

3.3.2. Via Langevin formalism

The original microscopic derivation of the deterministic DDFT for colloidal fluidsby Marini Bettolo Marconi and Tarazona [12, 13] starts from Langevin equations.Without hydrodynamic interactions and inertial terms, the motion of the i-th particleis governed by

d~ri(t)

dt= −Γ~∇~ri

( N∑j=1

j 6=i

U2(~ri − ~rj) + U1(~ri)

)+ ~χi(t), (72)

where ~ri(t) is the position of the i-th particle at time t, U2 the pair-interaction poten-tial, U1 an external potential, and ~χi(t) a noise term with the properties

〈~χi(t)〉 = ~0, (73)

〈~χi(t)⊗ ~χj(t′)〉 = 2D1δijδ(t− t′). (74)

After some manipulation using stochastic calculus, one finds the dynamic equation

∂

∂tρ(~r, t) = ~∇ · (

√2ΓkBT ρ(~r, t)~η(~r, t))

+ Γ~∇ ·(ρ(~r, t)

∫d3r′ ρ(~r′, t)~∇U2(~r − ~r′)

)+D~∇2ρ(~r, t)

(75)

with the microscopic density ρ given by Eq. (47) and the vector noise ~η that has zeromean and is δ-correlated in space and time. Equation (75) is statistically equivalentto Eq. (72) and identical to the result (48) derived by Dean [105].

The next step is to introduce the ensemble average as an average over all possiblerealizations of the noise, i.e., we consider an ensemble of microscopic states with thesame initial conditions, but different realizations of the noise. For the averaged density

24

ρ(~r, t) = 〈ρ(~r, t)〉, we obtain from Eq. (75) the deterministic equation

∂

∂tρ(~r, t) = ~∇ ·

(D~∇ρ(~r, t) + Γρ(~r, t)~∇U1(~r)

+ Γ

∫d3r′ 〈ρ(~r, t)ρ(~r′, t)〉 ~∇U2(~r − ~r′)

).

(76)

In the case of interacting particles, this is not a closed equation due to the presence ofthe two-point correlation function 〈ρ(~r, t)ρ(~r′, t)〉. A dynamic equation for this quan-tity would depend on the three-point correlation function. An infinite hierarchy ofcorrelation functions can be avoided by using approximations.

It is at this point that we make the central adiabatic approximation, which is usedhere to close the governing equation (76) for the ensemble-averaged one-body den-sity ρ(~r, t). The central idea is to approximate the time evolution by a sequence ofequilibrium states. In equilibrium, one can show that for each ρ(~r), there exists, givena fixed temperature, chemical potential, and interaction, a unique external potentialsuch that ρ(~r) is the equilibrium distribution (see Section 2.2). Thermodynamic equi-librium implies that [274]

1

ρ(~r)

∫d3r′ ρ(2)(~r, ~r′)~∇U2(~r − ~r′) = −kBT

∫d3r′ c(2)(~r, ~r′)~∇ρ(~r′) = ~∇δFexc[ρ]

δρ(~r)(77)

with the two-point density-density correlation (two-particle density) ρ(2)(~r, ~r′), thedirect pair-correlation function c(2)(~r, ~r′), and the equilibrium excess free energy func-tional Fexc[ρ]. We now make the approximation of replacing the nonequilibrium corre-lation 〈ρ(~r, t)ρ(~r′, t)〉 in Eq. (76) by the equilibrium correlation ρ(2)(~r, ~r′). By insertingEq. (77) into Eq. (76), we then obtain the DDFT equation

∂

∂tρ(~r, t) = Γ~∇ ·

(ρ(~r, t)~∇ δF [ρ]

δρ(~r, t)

)(78)

with the free energy (10).

3.3.3. Via Smoluchowski formalism

An alternative way for the derivation of DDFT is to start from a Smoluchowski equa-tion, as done by Archer and Evans [14]. Their derivation also includes multibody in-teractions. Neglecting hydrodynamic interactions, the Smoluchowski equation governsthe many-particle probability density Ψ as

∂

∂tΨ(~rk, t) = Γ

N∑i=1

~∇~ri · (kBT ~∇~ri + ~∇~riU(~rk, t))Ψ(~rk, t). (79)

The one-particle number density ρ(~r, t) is introduced as

ρ(~r1, t) = N

∫d3r2 · · ·

∫d3rN Ψ(~rk, t). (80)

25

We integrate in Eq. (79) over the coordinates of all particles except for one and write~r, ~r′, ~r′′, . . . instead of ~r1, ~r2, ~r3, . . . for consistency with Section 3.3.2. Using D =ΓkBT , this gives

∂

∂tρ(~r, t) = D~∇2ρ(~r, t) + Γ~∇ · (ρ(~r, t)~∇U1(~r, t))

+ Γ~∇ ·∫

d3r′ 〈ρ(~r, t)ρ(~r′, t)〉 ~∇U2(~r, ~r′)

+ Γ~∇ ·∫

d3r′∫

d3r′′ 〈ρ(~r, t)ρ(~r′, t)ρ(~r′′, t)〉 ~∇U3(~r, ~r′, ~r′′)

+ · · · ,

(81)

where, in contrast to Eq. (76), n-body-interaction potentials Un are included.Again, we face the problem that Eq. (81) is not a closed equation for ρ(~r, t), since

it involves the unknown n-body correlations. As in the derivation from the Langevinequations (see Section 3.3.2), a closure is possible using an adiabatic approximation:For an equilibrium fluid, we can relate the interparticle forces to the direct one-bodycorrelation function c(1)(~r) as

−kBTρ(~r)~∇c(1)(~r) =

∫d3r′ ρ(2)(~r, ~r′)~∇U2(~r, ~r′)

+

∫d3r′

∫d3r′′ ρ(3)(~r, ~r′, ~r′′)~∇U3(~r, ~r′, ~r′′) + · · · ,

(82)

where ρ(3) is the three-particle density, and c(1) to the excess free energy Fexc as

kBTc(1)(~r) = −δFexc[ρ]

δρ(~r). (83)

Making the approximation that these relations also hold out of equilibrium, which cor-responds to replacing the nonequilibrium correlations in Eq. (81) by their equilibriumcounterparts, we again find

∂

∂tρ(~r, t) = Γ~∇ ·

(ρ(~r, t)~∇ δF [~ρ]

δρ(~r, t)

). (84)

3.3.4. Via projection operator formalism

A third option for the derivation of DDFT is to use the Mori-Zwanzig projection op-erator formalism. This is a very general method for deriving theories for mean valuesand fluctuations of an arbitrary set of κ observables Ai(~z) (“relevant variables”)that depend on the phase-space coordinates ~z. (The formalism is also applicable toquantum systems, but we only require the classical case here.) The key idea is toproject the complete microscopic dynamics of a system onto a subset of relevant vari-ables Ai(~z). From the statistical point of view, this corresponds to replacing the

microscopic probability density Ψ by a relevant probability density ρ. The projectionoperator method exists in two forms. In the microcanonical framework or nonlinearLangevin theory [275], one fixes the precise values of the relevant variables [276]. Thisgives a Fokker-Planck equation for the probability distribution of the relevant vari-ables or, equivalently, a set of Langevin equations. In the canonical framework, on the

26

other hand, one fixes the mean values and projects the full microscopic probabilitydensity onto the corresponding local equilibrium distribution. A DDFT derived usinga microcanonical projection operator has a free energy that differs from that of DFT[277]. For deriving a DDFT with the free energy functional of DFT, one requires acanonical method [15]. Additional difficulties can arise from the fact that the relevantvariable of DDFT (the density operator (47)) is a sum over Dirac delta distributions[16,278]. A simple introduction to projection operators can be found in Ref. [37].

We start by explaining the canonical projection operator formalism introduced inRef. [36], which is a generalization of the method used by Espanol and Lowen [16]to derive DDFT. The projection operator P(t) acting on a dynamical variable Y isdefined as [279]

P(t)Y = Tr(ρ(t)Y ) +

κ∑i=1

(Ai − ai(t)) Tr

(∂ρ(t)

∂ai(t)Y

), (85)

where the relevant density has the grand-canonical form

ρ(t) =1

Ξ(t)exp

(− β

(HN (t)− µN −

κ∑i=1

a\i(t)Ai

))(86)

with the normalization

Ξ(t) = Tr

(exp

(− β

(HN (t)− µN −

κ∑i=1

a\i(t)Ai

))), (87)

time-dependent N -particle Hamiltonian HN (t), and thermodynamic conjugates

a\i(t) (the superscript \ denotes a thermodynamic conjugation), which ensure thatthe macroequivalence condition ai(t) = Tr(ρ(t)Ai) is satisfied, where ai(t) is the meanvalue of the variable Ai. Defining the coarse-grained free energy as

F (t) = Tr(ρ(t)HN (t)) + kBT Tr(ρ(t) ln(ρ(t))) (88)

allows to express the thermodynamic conjugates as

a\i(t) =∂F (t)

∂ai(t). (89)

If the relevant variable Ai is a field (as in DDFT), the partial derivative in Eq. (89)becomes a functional derivative. Frequently, one also needs the complementary oper-ator Q(t) = 1− P(t). One can then show that the time evolution of the mean valuesis given by [36]

ai(t) = vi(t)−κ∑j=1

∫ t

0dt′Rij(t, t′)βa

\j(t′) + zi(t, 0) (90)

with an overdot denoting a derivative with respect to time, the organized drift

vi(t) = Tr(ρ(t)iLSp(t)Ai), (91)

27

the (classical) retardation matrix

Rij(t, t′) = Tr(ρ(t′)(Q(t′)G(t′, t)iLSp(t)Ai)iLSp(t′)Aj), (92)

the orthogonal dynamics propagator

G(t′, t) = expR

(i

∫ t

t′dt′′ LSp(t′′)Q(t′′)

), (93)

the mean random force

zi(t, 0) = 〈Q(0)G(0, t)iLSp(t)Ai〉 , (94)

the imaginary unit i, the Schrodinger picture Liouvillian LSp(t), and the right-time-ordered exponential expR(·). Although it is less well known, a difference betweenHeisenberg picture and Schrodinger picture also exists for classical systems. A cor-responding discussion can be found in Refs. [36,37,280]. If the Hamiltonian dependson time, the Liouvillian in the Schrodinger picture differs from its form in the Heisen-berg picture, such that it is important to clarify which picture is used [36,37]. Themean random force (94) vanishes if the initial distribution is given by ρ(0), which weassume in this review. To see this, note that Eq. (94) can be written as

zi(t, 0) = Tr(Ψ(0)Q(0)G(0, t)iLSp(t)Ai)

= Tr(δΨ(0)G(0, t)iLSp(t)Ai),(95)

where δΨ(t) = Ψ(t) − ρ(t) [279,281]. To move from the first to the second line ofEq. (95), we have transposed Q in order to apply it to the phase-space distribution

Ψ. Although noise due to the irrelevant variables is, of course, present in general andcauses dissipation, the mean random force in Eq. (90) depends on whether the initialrelevant density ρ(0) is a good approximation for the actual distribution. This can bethought of as a condition for the construction of ρ [279], the form (86) is then mostappropriate if the system is initially in a state of constrained equilibrium [281].

From now on, we also assume time-independent Hamiltonians (see Refs. [36,282] forthe more general case). In this case, we can simply write L instead of LSp, since there isno difference between Schrodinger and Heisenberg picture Liouvillians. If the relevantvariables relax slowly compared to microscopic degrees of freedom, one can approx-imate the exact evolution equation (90) in the form (“Markovian approximation”)[16]

ai(t) = vi(t)−κ∑j=1

Dij(t)βa\j(t) (96)

with the diffusion tensor

Dij(t) =

∫ ∞0dt′ Tr(ρ(t)(Q(t)iLAj)e

iLt′(Q(t)iLAi)). (97)

This general theory allows to derive DDFT if the relevant variable is the one-particledensity (60), such that Eq. (96) provides a dynamic equation for ρ(~r, t) = 〈ρ(~r, t)〉.

28

(The fact that ρ(~r, t) is a field merely requires us to replace the sums in Eqs. (85),(86), (90), and (96) by integrals over space [37].) This was suggested by Kawasaki[115, 276]. Yoshimori [283] and Munakata [102] derived DDFT using the nonlinearLangevin equation method (see also Ref. [123]), which gives stochastic theories thatare not based on the DFT free energy. Finally, Yoshimori [15] and Espanol and Lowen[16] derived DDFT using canonical methods given by the Kawasaki-Gunton [284] andGrabert [279] projection operators, respectively. Projection operators also allow toderive generalizations of standard DDFT (see Section 5.3.1 for a discussion). For thisderivation method, the crucial step is the Markovian approximation that allows us tomove from Eq. (90) to Eq. (96). In the derivation of DDFT, which starts from theunderdamped Hamiltonian dynamics, it is this approximation that gives a memory-less overdamped dynamic equation. Physically, it is the assumption that ρ(~r, t) variesslowly on time scales on which the current correlation decays. This step leads to anoverdamped dynamics, since the momentum is assumed to relax quickly.

We here present the derivation by Espanol and Lowen [16], which is also discussed inRef. [37]. It is based on a grand-canonical framework, which is why it employs a density

functional Ω rather than a free energy F . This is justified in the thermodynamic limit.(Note that this is a property of this particular derivation and not a general restrictionof the projection operator method.) If the relevant variable A is the density ρ givenby Eq. (47), the grand-canonical partition function (87) becomes

Ξ(t) = Tr

(exp

(− β

(HN − µN − β

N∑i=1

ρ\(~ri, t)

)))(98)

with the thermodynamic conjugate ρ\. The partition function allows to introduce thegrand-canonical potential

Ω[ρ\] = −kBT ln(Ξ(t)). (99)

Based on the density functional Ω, introduced through the Legendre transformation

Ω[ρ] = Ω[ρ\] +

∫d3r ρ(~r, t)ρ\(~r, t), (100)

we can express the thermodynamic conjugate as

ρ\(~r, t) =δΩ[ρ]

δρ(~r, t). (101)

The organized drift vi(t) in Eq. (96) vanishes and the diffusion tensor (97) can beexpressed in terms of the currents

~J(~r) =

N∑i=1

~viδ(~r − ~ri) (102)

with the particle velocities ~vi as

Dij → (~∇~r ⊗ ~∇~r′) : D(~r, ~r′, t), (103)

29

where : is the double tensor contraction and D(~r, ~r′, t) is the tensor

D(~r, ~r′, t) =

∫ ∞0dt′ Tr

(ρ(t) ~J(~r′, 0)⊗ ~J(~r′, t′)

). (104)

Using Eqs. (96) and (101) and integration by parts, we obtain the dynamic equation[16]

∂

∂tρ(~r, t) =

1

kBT~∇~r ·

∫d3r′D(~r, ~r′, t)~∇~r′

δΩ[ρ]

δρ(~r′, t). (105)

To obtain the standard local equation (35), one assumes that positions and velocitiesare statistically independent and that positions evolve much slower than velocities.Both approximations are reasonable for a dilute colloidal suspension. In this case, thetensor (104) can be approximated as

D(~r, ~r′, t) ≈ D1ρ(~r, t)δ(~r − ~r′), (106)

where the diffusion constantD can be expressed in terms of the velocity autocorrelationfunction by the usual Green-Kubo expression. Inserting Eq. (106) into the general form(105) gives the standard result

∂

∂tρ(~r, t) =

D

kBT~∇ ·(ρ(~r, t)~∇ δΩ[ρ]

δρ(~r, t)

). (107)

For higher colloidal densities, more sophisticated approximations for the diffusiontensor can be developed that include hydrodynamic interactions. This includes, asdiscussed by Espanol and Lowen [16], the forms proposed in Refs. [41,133] (see Sec-tion 4.2).

The described derivation starts from the deterministic Hamiltonian dynamics ratherthan from the stochastic Langevin dynamics. Alternatively, a derivation of DDFT us-ing projection operators is possible starting from the Smoluchowski equation, providedthe projection operator method is adapted to stochastic microscopic dynamics [271].Since the procedure is also applicable to variables other than ρ(~r, t), it allows for theeasy derivation of more general theories involving other variables. Note that, althoughEq. (90) is formally exact, making a Markovian approximation here corresponds tothe assumption that the particle density is the only relevant variable, i.e., that othervariables such as the momentum relax quickly (overdamped limit). The projectionoperator formalism also allows to study the dynamics of fluctuations and correlationfunctions [36,281]. For DDFT, fluctuations were considered by Yoshimori [15] andcorrelation functions by Wittkowski et al. [40].

3.4. Accuracy

3.4.1. Limitations of DDFT

Although DDFT is, in general, a very successful theory, it has certain limitations. Theseresult from the approximations and assumptions involved in its derivation. Some ofthem can be avoided by using extensions of DDFT, which will be discussed in Section 4.

30

We mostly confine ourselves to deterministic DDFT in this section, since stochasticDDFT is exact for Brownian particles if it is formulated as a theory for the densityoperator (47). An obvious drawback of a stochastic DDFT for the microscopic densityis, of course, that it is impossible to solve in practice.

In general, DDFT calculations involve three approximations [39]: (i) using a grand-canonical rather than a canonical free energy, (ii) making an adiabatic approximation,and (iii) using approximate rather than exact grand-canonical functionals. A fourthproblem is that it allows hard particles to pass through each other [39,285].

Approximation (i) arises from the problem that DDFT is a dynamical theory thathas the form of a continuity equation which conserves the total number of particles.Hence, the canonical free energy functional F should be used. However, this is notknown, such that DDFT uses the grand-canonical functionals from DFT instead. Thiscan lead to wrong predictions for systems that only have a few particles or large densitygradients [38]. Consequences include unphysical self-interactions resulting from thecoupling to a grand-canonical reservoir [285]. Canonical approaches to DDFT havealso been developed (see Section 4.9).

Approximation (ii) arises from the fact that DDFT is a closed equation of motionfor the one-body density. In the microscopic derivation of DDFT (see Section 3.3.2),one arrives at Eq. (76), which is not closed as it also depends on the pair correlation.As a closure, one makes the adiabatic approximation. It states that the relaxation ofthe system is very slow such that it can be assumed that the pair correlation is alwaysgiven by that of the corresponding equilibrium system (which then allows to relate thepair correlation to the functional derivative of the equilibrium free energy). This is, ofcourse, not exactly true, such that predictions of DDFT for nonequilibrium pair corre-lations are not correct [286]. In other words, DDFT does not include “superadiabaticforces” [287] (see Section 5 for a discussion of these forces as well as of extensions ofDDFT including them). The assumption that a system is always in a local equilibriumstate often leads to relaxation times that are too short [277]. Similar problems arisein the modeling of dense nonequilibrium steady states, such as in channel flow [288].In particular, flow fields can induce distortions of the correlation function that DDFTdoes not capture [289]. Moreover, memory effects are not included [40].

The fact that in simple forms of DDFT the density is the only variable has anumber of consequences. A rather obvious one is that properties of the system thatare not captured by the density profile alone, such as temperature gradients, cannotbe described. A less obvious consequence, discussed in Section 4, is that by only usingthe density as a relevant variable, one effectively assumes that the other variablescharacterizing the system relax very quickly compared to the density, which is a strongapproximation [16,290]. For simple DDFT to be accurate, it is therefore required thatthe actual currents are diffusive rather than convective [291].

A related point is the fact that DDFT, being a dynamical theory for the one-bodydensity only, cannot distinguish between two states that have the same one-body den-sity but different pair-correlations. In equilibrium DFT, the one-body density gives, asdiscussed in Section 2.2, all relevant information about the system. For a system ini-tially out of equilibrium, however, this is not the case due to an additional dependenceon the initial condition [60,130]. Consequently, it is possible that a nonequilibrium sys-tem has an equilibrium density profile that minimizes the free energy functional, buta nonequilibrium correlation. Such problems can be relevant, e.g., for glassy systems[292].

Approximation (iii) is also related to the glass transition, since approximations forthe free energy can lead to the existence of local minima [12]. A poor approximation

31

can lead to the observation of a dynamic arrest that disappears when using a moreaccurate free energy [159,293]. Finally, the inability to describe a strict particle ordermakes it problematic to apply DDFT to nonergodic systems where caging is important[39]. Both aspects are discussed in Section 8.1.5.

3.4.2. Tests of DDFT

The success of DDFT results, of course, also from the fact that it has proven to bereliable. Usually, DDFT is tested by comparing its predictions to simulations of theunderlying (Brownian) dynamics for which it is an approximation.

In Brownian dynamics (BD) simulations, the microscopic equations of motion forthe individual particles are solved numerically. This allows to check whether the DDFTresults for the same system are correct. The outcomes of BD simulations can also beused to analyze where shortcomings of DDFT, if existing, come from, e.g., whetherthey arise from a bad choice of the free energy functional or from a breakdown ofthe adiabatic approximation [294]. They also allow to test the validity of assumptionsmade in DDFT and related theories regarding fluid properties such as the viscosity[295].

Comparisons between DDFT and BD simulations are made frequently. Examplesof studies combining DDFT and microscopic simulations consider active particles[20,296–300], advected DDFT [132], anisotropic pair correlations [286], Brownian ag-gregation [190,191], canonical dynamics [38], capillary collapse [301], capillary inter-actions [302], cluster crystallization [244], colloidal fluids in a cavity [272,303], col-loidal shaking [294], colloids in a DNA solution [304,305], demixing [306], diffusion-controlled reactions [307], drag forces [308], driven particles [18,19,309–311], dynam-ical correlations [312], dynamic mode locking [313], flowing colloids [314], hydrody-namic interactions [41,153,200,210,315–321], inhomogeneous polymer systems [322],lane formation [323–326], plasmas [327,328], power functional theory [25,30,32,329–332], protein-polyelectrolyte interaction [333], protein solvation [334], rheology [335],sedimentation [133,336,337], shear flow [289], solvation dynamics [338], stratification[188,339], superadiabatic forces [287,340], thermostatted granular fluids [341], the vanHove function [342,343], transport diffusivity [344], traveling band formation [345],ultrasoft colloids [17,346], and vacancy diffusion [347]. In Fig. 7 on page 95, a com-parison between DDFT results and BD simulations for a system of driven colloids isshown. The agreement is very good, which confirms the accuracy of DDFT.

Moreover, DDFT can be tested by comparing its predictions to experiments. Thiswas done for, e.g., active particles [348], Brownian hard disks [204], charging processes[349], colloids in a DNA solution [304,305,350], crystals [207,208], diffusion and hy-drodynamic interactions [315], dynamic mode locking [313], ion channels [351,352],nonequilibrium sedimentation of colloids [133,337], particles in confinement [194],phase separation [353], Poisson-Nernst-Planck (PNP) theory [354], protein adsorption[355,356], protein-polyelectrolyte interaction [333], resistance nonadditivity [357], thevan Hove function [358,359], and wetting [208]. Not discussed here due to lack of spaceis the large body of work that applies polymer DDFT (see Section 3.2.3) to experi-mental results and employs it in technological applications. Among the many examplesare Refs. [360,361] considering microphase separation in polymers, Ref. [55] discussingphase transitions in nanostructured fluids, and Ref. [362] studying the phase behaviorof thin films of block copolymers. Finally, an interesting opportunity for further testsof DDFT are experiments on diffusion in space [363].

A third option for testing DDFT is to compare it to exact results [12,273], provided

32

these are available.

3.5. Further aspects

3.5.1. Terminology

Terminological difficulties can arise since the name “DDFT” is not always used withthe same meaning in the literature, and since DDFT also has other names. Veryoften, “DDFT” is used to refer to the theory derived by Marini Bettolo Marconi andTarazona [12]. However, stochastic theories [114] (see Section 3.2.2) and methods forpolymers [10] (see Section 3.2.3) are also called “DDFT”. Some authors see phasefield crystal models (see Section 6.3) as a form of DDFT [364–369], while others donot [370]. Moreover, “dynamical density functional theory” is the name of a nucleationtheory proposed by Talanquer and Oxtoby [371]. Finally, the abbreviation “DDFT” isalso used in economics for “degree of dependence on foreign trade” [372] and in equineanatomy for “deep digital flexor tendon” [373,374].

On the other hand, various names are used for DDFT in the literature. The mostcommon ones are “dynamical density functional theory”, “dynamic density functionaltheory”, and “time-dependent density functional theory”. Note that the name “time-dependent density functional theory” is not only used for the classical DDFT, butalso for the quantum-mechanical method [81] and for lattice models [106]. More exoticnames for DDFT include “generalized diffusion approach” (GDA) [48] and “time-dependent classical density functional formalism” [375]. For stochastic theories, “time-dependent density functional method” [283], “stochastic density functional theory”[243], and “Dean-Kawasaki equation” [242] are alternatives to “DDFT” or “TDDFT”.The polymer DDFT by Fraaije and coworkers is referred to as “dynamic mean-fielddensity functional method” [124] or “dynamic density functional theory” [10].

33

Hamiltonian dynamics

Brownian dynamics

Stochastic DDFTfor microscopic density

Deterministic DDFTfor ensemble-averaged

density

Stochastic DDFTfor coarse-grained

density

MCT forcoarse-grained

density

Microscopic MCT

coarse graining,overdampeddynamics

statisticallyequivalent

MCT

approxim

ation

MCT

approxim

ation

coarse

graining,

approxim

ationforinteractions

reductionof

degrees

offreedom

,

adiabatic

approxim

ation

slow

den

sity,en

semble

averag

e

ensembleaverage,adiabaticapproxi-mation

coarse graining,approximationfor interaction

term

coarse

graining,

mom

entum

relaxes

fast

projectionop

erator

form

alism

momentumrelaxes fast

Figure 3. Relations of different forms of DDFT and MCT.

3.5.2. Deterministic or stochastic?

An occasional source of confusion and controversy is the question whether DDFTshould contain a noise term, i.e., whether Eq. (35) or Eq. (36) is correct [241]. In fact,they are both correct, but have a different meaning [211]. As discussed in Section 3,DDFT exists in two forms, as a deterministic and as a stochastic theory. Here, we willdiscuss how they are related. An illustration is given in Fig. 3. The crucial point here isthe interpretation of the density field ρ. While it always denotes an ensemble-averageddensity in static DFT, its meaning in DDFT can vary.

We start with the deterministic DDFT derived by Marini Bettolo Marconi andTarazona [12] (on the left in Fig. 3). Here, the density ρ(~r, t) is an ensemble average,i.e., an average over all possible realizations of the noise. Physically, these correspondto fixed initial positions of the colloids, but different microscopic configurations ofthe bath (“Brownian ensemble”) [12]. This average, given an initial state, is uniquely

34

determined. Hence, there can be no noise terms [14,211]. These would lead to the wrongequilibrium limit, since F [ρ] already includes the fluctuations and adding them to thetime evolution would correspond to overcounting them [139]. In deterministic DDFT,the free energy functional F has the same form as in DFT. This is the main advantageof this form, since, in constructing the free energy functional, one can make use of thewell-developed theory of DFT [12]. Some examples are discussed in Section 2.2.2.

There is, however, a second aspect that has to be taken into account: The exactdescription always gives a convex thermodynamic potential, such that through thedynamics the free energy approaches its unique (local and global) minimum. In prac-tice, one will use approximate free energy functionals, which might have more thanone local minimum. In this case, it is possible that through the time evolution thesystem gets trapped in a metastable state corresponding to a local minimum, suchthat phenomenological noise terms have to be added to push the system out of themetastable states towards its true equilibrium [12,139].

One can, however, also define ρ(~r, t) as a spatially or temporally coarse-grained den-sity (on the right in Fig. 3 on the previous page). In this case, the equations of motionfor ρ(~r, t) are stochastic [14,211]. The density ρ(~r, t) appearing in the deterministicequation can be interpreted as the probability density of finding a particle at time t atposition ~r, whereas the density appearing in the stochastic equation is the number ofparticles in a small volume [278,376]. Coarse graining in the form of a spatial average,introduced by dividing space into small cells, is employed in the DDFT by Kawasaki[114] (see Section 3.2.2).

A further possibility for averaging in addition to noise, space, and time is an averageover initial conditions. This possibility is discussed in Refs. [238,316]. Averaging overinitial conditions leads to a smooth initial density profile ρ(~r, 0) instead of a “spiky”distribution that has peaks at the positions of the particles. The time-dependent den-sity profile ρ(~r, t) is a functional of the initial density profile and is thus different ifsuch an average is taken [316]. This aspect can be relevant for phase separation, asdiscussed in Section 8.2.2. One can combine this type of averaging with various typesof DDFT. For example, Bleibel et al. [316] employ it in the context of deterministicDDFT to study capillary collapse, while Donev and Vanden-Eijnden [238] discuss itspossible application to a stochastic model.

Finally, a DDFT can be formulated as a theory for the density operator ρ(~r, t) givenby Eq. (47), which is a sum of Dirac delta distributions (in the middle of Fig. 3 on thepreceding page). This DDFT provides an exact description of a system of Brownianparticles (a paradigmatic example is the result (50) obtained by Dean [105]), or anapproximate description of a system of Hamiltonian particles. A DDFT for the densityoperator is also stochastic. Note that the free energy entering Dean’s DDFT is given byEq. (49) and not by the DFT free energy [211]. To emphasize this difference, the freeenergy (49) is sometimes denoted as H[ρ] rather than F [ρ]. Dean’s stochastic DDFTcan serve as a basis for the derivation of a deterministic DDFT based on ensembleaverages and further approximations [12] (see Section 3.3.2). Although it is exact, itmight be insensitive to the conservation of particle order in 1D as it uses a symmetricpair-interaction potential (an effect described by Schindler et al. [39] for deterministicDDFT, see also Section 8.1.5).

A detailed discussion of the relation between deterministic and stochastic DDFTwas given by Archer and Rauscher [211] (see also Refs. [70,143]).

Another way to understand the differences between deterministic and stochasticDDFT is to compare their microscopic foundations in the projection operator formal-ism (see Section 5.3.1). This method allows for a systematic derivation of field theories,

35

such as DDFT, by projecting the microscopic dynamics onto the subdynamics of rel-evant variables. For the projection operator method, two forms can be distinguished,called “microcanonical” and “canonical” form [276,277]. In the microcanonical form,also known as “Zwanzig projection operator” or “nonlinear Langevin theory”, oneprojects onto states with fixed values of the relevant variables, i.e., the density ρ(~r, t)is specified exactly. The free energy functional one gets is the “microcanonical free en-ergy”. This name is chosen in analogy to the microcanonical ensemble, where, unlikein the canonical ensemble, one specifies the exact value of the energy rather than themean value. Microcanonical methods lead to stochastic theories, where the equationsderived in the projection operator framework contain a random noise term. This is nota problem in the equilibrium limit, since the free energy appearing in stochastic DDFTis different from the free energy of deterministic DDFT. In the microscopic derivationof stochastic DDFT, one gets a bare free energy and bare transport coefficients [278].These can be connected to the renormalized free energy of DFT and the renormalizedtransport coefficients of deterministic DDFT through a fluctuation renormalization[277]. This procedure is discussed in Ref. [279].

On the other hand, in a canonical theory, such as the “Robertson projection opera-tor method” [377] or “Kawasaki-Gunton projection operator method” [284], one fixesonly the mean values of the relevant variables (such as the one-body density) ratherthan the exact values. This is analogous to the canonical ensemble in statistical me-chanics, where fluctuations of the energy around its mean value are allowed. Here, thefree energy is of the canonical form, which is the one appearing in DFT [15]. Hence,fluctuations are already included in the free energy functional. From this approach,one can derive the well-known form of deterministic DDFT [16]. (Formally, neglect-ing the noise term in the averaged equations requires an additional assumption aboutinitial conditions [15,37].) It is also possible to interpolate between deterministic andstochastic theories using more general projection operators [277]. A further discussionof the relation of deterministic and stochastic methods based on projection operatorscan be found in Ref. [278], where the problem is also related to numerical aspects.

Similar considerations as for DDFT apply to phase field crystal models [290,378–380] and Cahn-Hilliard models [381].

3.5.3. Role of the solvent

In the context of versions and applications of DDFT, it is important to pay attention tothe difference between “atomic/molecular” and “colloidal” fluids. A (one-component)atomic/molecular fluid (henceforth called a “simple” fluid) consists of a large numberof identical particles (atoms or molecules) that interact through a certain two-bodyinteraction potential. These interactions are governed by the deterministic laws ofNewtonian mechanics. Coarse graining the microscopic description of fluids leads to thewell-known equations of hydrodynamics. A colloidal fluid, on the other hand, consistsof small “solvent particles” in which much larger “colloidal particles” are immersed.The exact microscopic description of colloidal fluids is challenging because, due tothe huge difference in mass and size between the two particle types, very differenttime scales have to be considered. Typically, one models only the colloidal particlesand describes their interaction using an effective theory. The solvent particles enterthe dynamics through the noise term and the friction force. This gives a descriptionin terms of Langevin equations or, equivalently, a Smoluchowski equation. Since notall microscopic degrees of freedom are considered on the colloid level of description,these equations are stochastic. The Smoluchowski equation can be derived by coarse

36

graining from the exact microscopic Liouville equation that describes both solventand colloidal particles [272]. A useful method for connecting the mesoscopic dynamicsof particles in a solvent to the microscopic level is the projection operator formalism(see Section 5.3.1), as shown by Espanol and Donev [382]. Duran-Olivencia et al. [383]obtained the coarse-grained description of the N -body distribution for the colloidsfrom the full microscopic dynamics using the projection operator method.

The original microscopic derivations of deterministic DDFT, based on Langevinequations [12] and Smoluchowski equation [14], started from the overdamped stochasticdynamics and thus apply to colloidal fluids. In these theories, the density describes thenumber of colloidal particles per unit volume and not, as in hydrodynamics, the massdensity. This is different in DDFT-based descriptions of atomic fluids. A DDFT foratomic fluids, which has as a microscopic starting point Newton’s equation of motionrather than Langevin equations, was derived by Archer [131]. The resulting dynamicequation (148) includes inertial effects and is a differential equation of second orderin time derivatives. In the limit of large collision frequencies, the usual first-orderoverdamped DDFT equation (35) is recovered. For the description in the projectionoperator formalism, the situation is more subtle. Espanol and Lowen [16] derived themore general nonlocal expression (105), which only contains the number density asa relevant variable (and thus assumes all other variables to relax quickly). For thecase of colloidal fluids, further approximations lead to the usual local form. A DDFTfor simple fluids was derived with the projection operator formalism by Camargoet al. [384]. Here, the momentum density is also used as a relevant variable. Moreover,standard DDFT for colloidal fluids can be recovered from a microscopic hydrodynamictheory for a binary mixture if one particle species (the colloids) is much heavier andmore dilute than the other one (the solvent) [385].

A further interesting topic is the dynamics of the solvent itself. Often, it is assumedthat the solvent relaxes very rapidly compared to the colloids. On the other hand,nonequilibrium effects arise if the relaxation time scales are comparable, in particularif confinement prohibits easy relaxation of the solvent [386,387]. Solvation dynam-ics (rearrangement of solvent particles when a solute particle is present) can also bemodeled in DDFT [386,388].

3.5.4. Approach to equilibrium

Among the central topics of nonequilibrium statistical mechanics is the question howand why systems approach a state of thermodynamic equilibrium [60,389]. The ten-dency to approach equilibrium is often associated with the second law of thermody-namics (see Ref. [390] for a detailed discussion) and has even been granted the statusof a “minus first law of thermodynamics” by Brown and Uffink [391]. An early andimportant result in this area is the H-theorem derived by Boltzmann [89]. It statesthat a quantity known as the H-functional, which depends on the phase-space distri-bution and is identified with the negative entropy (“negentropy”), is monotonouslydecreasing [90]. This has inspired a significant amount of work aiming at the deriva-tion of other H-theorems. DDFT, which describes the way systems approach a stateof thermodynamic equilibrium as described by DFT, is no exception.

In deterministic DDFT, one can prove that the free energy is monotonously de-creasing. This was done for the standard DDFT (35) by Munakata [101] and MariniBettolo Marconi and Tarazona [12] and for the generalized form (105) by Espanol and

37

Lowen [16]. From Eq. (35), one obtains

dF [ρ]

dt= −Γ

∫d3r ρ(~r, t)

(~∇ δF [ρ]

δρ(~r, t)

)2

≤ 0. (108)

It can be shown from microscopic calculations that the mobility Γ is positive, whichcompletes the proof of this “H-theorem” of DDFT [16]. The minimum of the freeenergy is reached for δF/δρ = µ with a constant chemical potential µ [101], which isin agreement with the variational principle (8) of static DFT. Hence, Eq. (35) indeeddescribes a monotonous approach towards the equilibrium state described by DFT.The precise speed of this relaxation is slightly overestimated due to the approximationsmade in the derivation of deterministic DDFT [277]. An interesting consequence of theresult (108) is that the time evolution can be trapped in local minima of the free energythat arise from approximations made in its construction [12]. This can be avoided byadding noise terms. A H-theorem for stochastic DDFT was derived by Munakata[101]. H-theorems can also be obtained for extended theories describing phase-spacedynamics [392,393].

Having discussed how systems approach equilibrium in DDFT, we can also addresswhy this is the case. A derivation of irreversible macrodynamics from reversible micro-dynamics requires a coarse-graining procedure [394]. The discussion of thermodynamicirreversibility has become a matter of intense discussion in philosophy of physics. Thisdebate is concerned with (a) the location of irreversibility within thermodynamics, (b)the definition of “thermodynamic equilibrium”, (c) the justification of coarse graining,(d) the approach to equilibrium, and (e) the direction of the arrow of time [60]. Beinga paradigmatic example of a theory describing the irreversible approach to equilibriumbased on a microscopic theory, DDFT can provide interesting insights into this debate.For example, the derivation by Espanol and Lowen [16] (see Section 3.3.4) starts fromHamilton’s equations, which are reversible, and obtains from them the DDFT equation(107), which is irreversible. This is a consequence of coarse graining combined with aMarkovian approximation [37]. A discussion of philosophical aspects of DDFT can befound in Ref. [60].

4. Extensions of standard DDFT

In this section, we discuss extensions of standard DDFT. By an “extension of standardDDFT”, we denote any form of DDFT that allows to model aspects not covered bythe theories presented in Section 3 (deterministic and stochastic DDFT for simple andcolloidal fluids and polymer DDFT). Such extensions exist in a large variety of forms.An overview is given in Table 1. As visualized in Fig. 4 on page 40, these extensions canbe distinguished by the way in which they modify the standard DDFT that is obtainedby deriving an approximate transport equation for the one-body density ρ(~r, t) basedon the microscopic dynamics of simple or colloidal fluids. Some extensions modifythe microscopic dynamics, e.g., by replacing the Langevin equations for passive bythose for active Brownian particles (see Section 4.6.3) or by considering quantumrather than classical systems (see Section 4.11). Other extensions reduce the numberof approximations in the derivation of the transport equation for ρ(~r, t), e.g., by notmaking an adiabatic approximation (see Section 5.2) or by not using an approximategrand-canonical free energy functional (see Section 4.9). This type of extensions alsoinvolves formally exact approaches, such as power functional theory (PFT) [26], which

38

are discussed in Section 5.Finally, one can use an extended set of order parameters rather than just the den-

sity field ρ(~r, t). Some examples which we present in Section 5 are extended dynamicaldensity functional theory (EDDFT) involving also the energy or entropy density, func-tional thermodynamics (FTD) involving also the energy density, and PFT involvingalso density currents or velocity gradients. Such extensions have two advantages. Thefirst and obvious one is that by considering an additional field (e.g., the energy den-sity) one has more detailed information about the system. A second and less obviouspoint is the range of applicability of the resulting theories. As an example, we take aderivation in the projection operator formalism (see Section 3.3.4): For a given set ofrelevant variables, one first rewrites the microscopic equations in an exact form, whichis possible regardless of how they are chosen. In the second step, which is crucial for thedynamic equations to be useful, one makes a Markovian approximation correspondingto the assumption that the relevant variables are slow compared to the degrees offreedom that are projected out. If we now compare a DDFT with and without energydensity as a relevant variable, the second one requires for a Markovian approximationthat all irrelevant variables – including the energy density – relax quickly on relevanttime scales, which for the energy means that the conjugate variable, which is the tem-perature, has equilibrated. In summary: A DDFT in which the number density is theonly variable effectively needs to assume that the temperature is constant, while aDDFT with energy density does not. A slightly different form of this extension is toextend the configuration space on which the density is defined, such as by using adensity %(~r, ~u, t) rather than ρ(~r, t) to incorporate orientational degrees of freedom ~u(see Section 4.6.1) or by including a momentum-dependence (see Section 4.7.3). Manyextensions combine a variety of these aspects.

In the following, we present a variety of extensions in (very roughly) increasingorder of complexity. A simple modification of the standard DDFT equation (35) isto make the diffusion coefficient nonconstant (Section 4.1), which includes the caseof hydrodynamic interactions (Section 4.2). More complex extensions involve addingadditional terms to Eq. (35), such as source and sink terms (Section 4.3) or advectionterms (Section 4.4). In addition, one can introduce additional order parameter fields,which in the simplest case are also density fields (Section 4.5). Further possibilitiesfor order parameter fields arise from additional degrees of freedom of the particles,such as orientation (Section 4.6) – giving rise to polarization and nematic order – andmomentum (Section 4.7) – giving rise to the momentum density. The latter extensionnaturally leads to the classical equations of hydrodynamics, which often also involvethe energy density (Section 4.8). Finally, we turn to special types of extensions thatinvolve an alternative approach to the construction of the free energy (Section 4.9)or are designed for specific applications such as polymer (Section 4.10) and quantum(Section 4.11) systems.

39

Microscopic dynamics(Hamiltonian, Langevin)

Other or more generalmicroscopic dynamics:

• Active DDFT• Quantum DDFT• . . .

Larger configu-ration space:

• Nonsphericalparticles(%(~r, ~u, t))

• Kinetictheory(Ψ(~r, ~p, t))

• . . .

Additional variables:

• Extended DDFT(mixtures,energy density)

• Functionalthermodynamics(energy density)

• DDFT withmomentumdensity

• . . .

dynamic equation for ρ(~r, t)approximations

(adiabatic closure,approximate

free energy, . . . )

Omission of certainapproximations:

• Particle-conser-ving dynamics(free energy)

• Power functionaltheory (adiabaticapproximation)

• . . .

Stepsin

derivationof

DDFT

Correspon

dingextension

s

Figure 4. Classification of extensions of standard DDFT.

40

Extension ReferencesNonconstant diffusion coefficient [22,40,41,70,153,189,193,200,209,210,214,215,

236,238,315–321,344,395–411]Hydrodynamic interactions [16,22,40,41,133,153,189,200,209,210,214,215,

238,315–321,396–411]Source and sink terms [46,49,50,52,412–420]Potential flow [22,132,288,304,396,397,409–411,421–426]Shear flow [19,24,26,51,137,182,224,225,256,289,329–

331,405,411,427–447]Mixtures [11,27,39,40,46,49–53,159,184,185,188,192,201,

203,204,210,272,285,286,293,294,303,306,317,324,325,328,332,333,339,342,343,349,353,354,356,358,380,385,411,414–420,448–485]

Nonspherical particles [20,21,88,109–112,121,175,176,181,183,239,334,358,359,383,401,486–494]

Magnetic particles [448–450]Active particles [20–23,32,205,296–300,411,418,461,493,495–498]Inertia [29,42,130,131,317,319,320,393,400,401,499–

503]Momentum density [47,88,143,186,257–259,317,327,328,385,402,

414,415,428,451,471,477,478,498,502,504–519]Kinetic theory [143,186,259,341,385,392,470,471,477,481,482,

499,500,502,505]Fixed temperature gradients [134,500]Energy density [40,138,186,212,385,505,510,520–522]Entropy density [212,520,522]Particle-conserving dynamics [38,39,273,523]Extensions of polymer DDFT [126,268,524,525]Quantum mechanics [28,47,88,512,513,526]

Exact generalization ReferencesPower functional theory [26–32,296,300,329,331,332,526–529]Projection operator formalism [15,16,33–37,40,102,114,138,212,271,276–

279,281–283,377,382–384,392,401,428,469,509,511,530–538]

Runge-Gross theorem [81,130]

Table 1. Overview over extensions and exact generalizations of standard DDFT.

41

4.1. Nonconstant diffusion coefficient

While in standard DDFT, the diffusion coefficient D is assumed to be a constant (andused as an input parameter), extended approaches allow it to vary. The most importantcase is the one of hydrodynamic interactions presented in Section 4.2. More generally,derivations in the projection operator formalism (see Sections 3.3.4 and 5.3.1) giveprecise microscopic expressions for the diffusion coefficient that can, in certain cases,be approximated by a constant. Liu [344] calculated (nonconstant) diffusion coeffi-cients based on an entropy scaling rule and applied this method to gas diffusion inporous media. A general discussion of density-dependent diffusion coefficients can befound in Ref. [395]. Moreover, a position-dependent diffusion coefficient can describean inhomogeneous environment, e.g., in transport through membranes [193]. Finally,the generalized Smoluchowski equation (46) discussed in Section 3.2.2 has a modifiedmobility function Γg(ρ) instead of the standard form Γρ [70,236].

4.2. Hydrodynamic interactions

Hydrodynamic interactions are the most important reason for modifying the diffusioncoefficient. A DDFT with hydrodynamics is a theory in which the mobility includeshydrodynamic interactions between the particles. In this case, solvent-mediated inter-actions lead to additional terms in the DDFT current. This results in a configuration-dependent diffusion tensor. Royall et al. [133] proposed a first simple treatment ofhydrodynamic interactions in DDFT, using an equation by Hayakawa and Ichiki [539]for the dependence of the mobility Γ on the colloid packing fraction. A microscopictheory including hydrodynamic interactions was derived by Rex and Lowen [41] (seealso Refs. [153,200,214,215]). In the Smoluchowski equation (in the simplest case givenby Eq. (79)), they used a two-particle diffusion tensor

Dij(~rk) ≈ D1δij +D

(δij

N∑l=1l 6=i

ω11(~ri − ~rl) + (1− δij)ω12(~ri − ~rj))

(109)

with the single-particle diffusion constant D. The tensors ω11 (self term) and ω12

(distinct hydrodynamic tensor) are given by [540]

ω11(~ri − ~rj) = As(‖~ri − ~rj‖)(~ri − ~rj)⊗ (~ri − ~rj)

‖~ri − ~rj‖2

+Bs(‖~ri − ~rj‖)(1− (~ri − ~rj)⊗ (~ri − ~rj)

‖~ri − ~rj‖2), (110)

ω12(~ri − ~rj) = Ac(‖~ri − ~rj‖)(~ri − ~rj)⊗ (~ri − ~rj)

‖~ri − ~rj‖2

+Bc(‖~ri − ~rj‖)(1− (~ri − ~rj)⊗ (~ri − ~rj)

‖~ri − ~rj‖2)

(111)

with the self-mobility functions As and Bs and the cross-mobility functions Ac andBc. On the Rotne-Prager level of approximation [541], these are given by [540]

As(‖~ri − ~rj‖) = O((

R

‖~ri − ~rj‖

)4), (112)

42

Bs(‖~ri − ~rj‖) = O((

R

‖~ri − ~rj‖

)4), (113)

Ac(‖~ri − ~rj‖) =3

2

R

‖~ri − ~rj‖−(

R

‖~ri − ~rj‖

)3

+O((

R

‖~ri − ~rj‖

)4), (114)

Bc(‖~ri − ~rj‖) =3

4

R

‖~ri − ~rj‖+

1

2

(R

‖~ri − ~rj‖

)3

+O((

R

‖~ri − ~rj‖

)4), (115)

where R is the radius of the particles (which are assumed to be spherical). If the hydro-dynamic radius of the particles differs from the interaction radius, one has to use thehydrodynamic radius (see Ref. [540]). This was done in Refs. [41,200]. The expressionfor Dij is valid for particles that are far away from each other. Using the method ofreflections [540,542], the mobility functions can be calculated up to arbitrary order.By integrating over the Smoluchowski equation, similar to the procedure employed forstandard DDFT (see Section 3.3.3), Rex and Lowen obtained the result

kBT

D

∂

∂tρ(~r, t) = ~∇~r ·

(ρ(~r, t)~∇~r

δF [ρ]

δρ(~r, t)

+

∫d3r′ ρ(2)(~r, ~r′, t)ω11(~r − ~r′) · ~∇~r

δF [ρ]

δρ(~r, t)

+

∫d3r′ ρ(2)(~r, ~r′, t)ω12(~r − ~r′) · ~∇~r′

δF [ρ]

δρ(~r′, t)

),

(116)

which, in addition to the standard current of DDFT (first term on the right-hand side),also involves a current from the reflection of solvent flow by the thermodynamic forceat position ~r (second term on the right-hand side) and from the solvent flow induced bythe thermodynamic force at position ~r′ (last term). The equation can be closed by usingan Ornstein-Zernike relation for the two-particle density ρ(2)(~r, ~r′, t) [41]. A simplifiedform of Eq. (116), which uses the Oseen tensor for the hydrodynamic interactions andan ideal gas correlation, is presented in Ref. [315]. The mobility can get an explicitspatial dependence if the translational symmetry of the system is broken by interfaces.In this way, hydrodynamic interactions with walls can be described [396,397].

Work on DDFTs including hydrodynamics has been performed in various directions.Derivations based on the projection operator formalism [16,40] naturally lead to dy-namic equations with hydrodynamic interactions (as discussed in Section 3.3.4), suchthat they are included in EDDFT [40] (see Paragraph 5.3.2.1). A rigorous derivation ofa Smoluchowski equation for the one-particle distribution incorporating hydrodynamicinteractions is given in Ref. [398]. Goddard et al. [399] examined the well-posednessof hydrodynamic DDFT. The inertia of particles is included in hydrodynamic DDFTsin Refs. [317,319,320,400–402]. Donev and Vanden-Eijnden [238] obtained a hydrody-namic DDFT that also includes the fluctuations (averaging then recovers the resultfrom Rex and Lowen [41]). Duran-Olivencia et al. [401] considered orientable particleswith arbitrary shape. Another extension is the study of hydrodynamic interactionsin multi-species systems [40,317]. A DDFT for polymer chains with hydrodynamicinteractions is presented in Refs. [210,403]. In Ref. [404], hydrodynamic interactionsin a flowing hard-sphere suspension are studied using Monte Carlo simulations. Ex-perimental results on the importance of hydrodynamic effects on short-time diffusion(which are compared to a DDFT without hydrodynamic interactions) can be found inRef. [204].

43

Regarding applications, the original hydrodynamic DDFTs were used to studysedimentation [133] and oscillations in optical traps [41]. Effects of hydrodynamicsin confined systems have also been studied in DDFT [315,316,318,321,405]. More-over, hydrodynamic DDFT allows to study hydrodynamic interactions in ion diffu-sion [406], wavenumber-dependent diffusion coefficients [316], crystal growth [189,407],glassy water [209], and velocities of particles subject to hydrodynamic stress [408]. Fi-nally, hydrodynamic interactions are included in DDFT descriptions of microswimmers[22,409–411].

4.3. Source and sink terms

A further possibility for modifying the standard DDFT equation (35) is to add sourceand sink terms. Sources and sinks can be a result from influx or outflux, but in thecase of mixtures (see Section 4.5) also originate from a transformation of one particletype into another. If the system of particles is not closed, the number of particles isnot necessarily conserved and particle inflow is possible. This was discussed by Lowenand Heinen [412] for the case of particles being injected into a confined system atposition ~r with rate qin(t). In this case, a point source qin(t)δ(~r) has to be added tothe DDFT equation (35). If Eq. (35) is linearized (assuming small density variations),an analytical solution of this problem is possible using Green’s functions. In Ref. [413],source terms are employed to describe diffusion of chemicals around a point source.Other cases in which source terms have to be added to the conserved DDFT equationinclude evaporating thin films [457] (see Section 8.1.4), surface charge densities [414,415] (see Section 8.1.10), biological dynamics [46,49] (see Section 8.1.11), chemicalreactions [52,416,417] (see Section 8.2.5), switching [418,419], and epidemic spreading[50,420] (see Section 8.2.6).

4.4. Flow

4.4.1. Potential flow

In general, it cannot be assumed that the solvent in which the colloids are immersedis at rest. Thus, a natural generalization of DDFT is to allow for solvent flow witha velocity ~v(~r, t). An additional modification of DDFT is the incorporation of shearflow, which will be discussed separately (see Section 4.4.2).

Early models studied effects, such as colloids being dragged through a polymersolution, by shifts to a frame that is co-moving with a velocity ~vcm [18] and did notconsider perturbations of the flow [543]. Essentially, this corresponds to assumingthat the solvent simply flows through the colloid [132]. However, as discussed in Refs.[132,288], the colloids also influence the solvent. If a spherical obstacle of radius Ro,such as a colloid, moves through a solution, the solvent particles, which are very small,can get very close. This leads to a modified flow field, which can for incompressibleflow at low Reynolds number be calculated analytically as

~v(~r) =3Ro

4r

(1 +

R2o

3r2

)~vcm +

3Ro

4r3(~r · ~vcm)~r

(1− R2

o

r2

)− ~vcm (117)

with r = ‖~r‖. Only for r →∞, Eq. (117) can be approximated by ~v = −~vcm.A DDFT for colloidal particles advected by a flow field was derived by Rauscher

44

et al. [132], who did not take hydrodynamic interactions between solvent and colloidalparticles into account. The starting point are the overdamped Langevin equations forthe motion of the i-th particle in the presence of a flow field ~v(~r, t), given by

d~ri(t)

dt= ~v(~ri, t)− Γ~∇~ri

(U1(~ri) +

N∑j=1j 6=i

U2(‖~ri − ~rj‖))

+ ~χi(t). (118)

For these Langevin equations, one can obtain a corresponding Fokker-Planck equationfrom which the evolution of the averaged density ρ is computed. If we have a potentialflow, which is always the case when detailed balance holds, we can write

~v(~r) = −Γ~∇Uvel(~r) (119)

and define a modified external potential U?(~r) = U1(~r) + Uvel(~r) with the velocitypotential Uvel. This allows to approximate the interaction term in the usual way. Theresulting advected DDFT reads

∂

∂tρ(~r, t) + ~∇ · (~v(~r, t)ρ(~r, t)) = Γ~∇ ·

(ρ(~r, t)~∇ δF [ρ]

δρ(~r, t)

). (120)

As can be seen, this corresponds to a standard DDFT equation with U1 replaced byU?.

In Ref. [396], this treatment is extended to hydrodynamic interactions with walls,incorporated by a position-dependent mobility, and hydrodynamic interactions amongthe particles (see Section 4.2). Praetorius and Voigt [421] used the advected DDFTto derive an advected PFC equation. A highly simplified model is applied to colloidsin a DNA solution in Ref. [304]. Moreover, advected DDFT is used in microrheology[288,422–424] (see also Section 8.2.2). A possible setup is the flow of interacting par-ticles around a fixed probe particle [424,425]. Hydrodynamic lift forces are discussedin Refs. [397,426]. Microscopic considerations of the flow field are also important inthe derivation of a hydrodynamic DDFT for microswimmers [22,409–411] (see Sec-tion 4.6.3). Problems of the local equilibrium closure are considered in Ref. [288]. Areview on nanoparticle flow can be found in Ref. [544].

4.4.2. Shear flow

Investigating the properties of sheared simple and complex fluids is a central topicin soft matter physics. Therefore, effects of shear were considered already at earlystages in the development of DDFT. The shear viscosity was calculated based onshear-induced deviations from the equilibrium profile [224,225]. Soft colloids shearedby traveling waves were studied by Rex et al. [19].

As discussed in Ref. [137], there exist certain problems in describing shear flowusing standard DDFT. To see this, we try to apply the governing equation of advectedDDFT (120) to a simple shear flow experiment with two parallel plane walls, where yis the direction normal and x the direction parallel to the walls. The flow velocity ~vhas the form

~v(~r) = yγ~ex (121)

45

with the shear rate γ and the unit vector in x direction ~ex. The density will, becauseof translational symmetry, only depend on y. Unfortunately, the form (121) implies

~∇ · (~v(~r)ρ(~r, t)) = 0, (122)

such that Eq. (120) reduces to the standard DDFT equation without a flow field.However, it is certainly wrong that shear has no effects on the particle distribution.The source of the problem is the distortion of the correlation functions (which DDFTassumes to be as in equilibrium) due to the external flow [289]. A DDFT-based de-scription of distorted correlation functions for nonlinear rheology is proposed in Ref.[335].

A solution to this problem has been suggested by Brader and Kruger [137]. Theflow velocity is corrected by the mean-field term

~vfl(~r, t) =

∫d3r′ ρ(~r′, t) ~Ks(~r − ~r′), (123)

where ~Ks(~r − ~r′) is the flow kernel accounting for the effects of shear flow. Whilethis is a phenomenological correction, systematic derivations are possible in certaincases [289]. A detailed discussion of shear DDFT can be found in Ref. [182] (see alsoRef. [405] for the role of hydrodynamic interactions in shear flow). A review article ontime-dependent shear flow can be found in Ref. [427].

Shear was studied in a variety of forms and extensions of DDFT. Since superadia-batic effects are important in rheology, it is reasonable to describe shear flow by PFT[26,330]. Superadiabatic forces in sheared systems are discussed in the framework ofPFT in Refs. [329,331,528,529]. Kruger et al. [256] derived the stress tensor for stochas-tic DDFT, considering the shear viscosity and shear-stress fluctuations. Shear can alsobe described within DDFT extensions including the momentum density [428].

An application of DDFT with shear-flow corrections is the study of the influenceof time-dependent shear on sedimentation [429]. In Ref. [430], shear DDFT is usedto analyze the interplay between shear and crystallization. A linear stability analysisallows to determine the dispersion relation for the laning instability [431]. Effects ofshear flow on nonequilibrium phase transitions are considered in Ref. [202]. Shear flowcan also have effects on the formation of low-density regions behind tracer particles[432]. Brader and Schmidt [24] discussed effects of shear flow on the time-dependentdiffusion tensor. Scacchi et al. [447] studied the influence of shear on length scales insoft matter. Polymer DDFT (see Section 3.2.3) is also frequently applied to shearedsystems [51,433–446]. Hoell et al. [411] model a shear cell consisting of microswim-mers trapped in passive colloidal particles. Note that not all applications of DDFTmentioned in this section employ the method by Brader and Kruger [137].

4.5. Mixtures

DDFT has, very frequently, been applied to systems consisting of different particletypes. The general idea, already employed by Munakata [11], is straightforward: Oneuses a separate density field ρi(~r, t) for each particle species i, defines a free energyF [ρi] that depends on all fields and includes the single-species dynamics as well aspossible interactions, and then writes down the DDFT equation (35) for each particle

46

type. The result is

∂

∂tρi(~r, t) = Γi~∇ ·

(ρi(~r, t)~∇

δF [ρj]δρi(~r, t)

). (124)

Here, Γi is the mobility of particle type i. Equation (124) is a special case of Eq. (216),which is the general form of a diffusive transport equation for a conserved order param-eter known as “gradient dynamics” (this can be seen by writing Eq. (124) in vectornotation by introducing a vector ~ρ whose elements are the fields ρi). To obtainEq. (124) from Eq. (216) for the case that the general conserved variable ac fromEq. (216) is given by ac = ~ρ, we have to assume that the mobility matrix Mg ap-pearing in Eq. (216) has the elements (Mg)ij = Γiρiδij (no sum over i), i.e., that alloff-diagonal coefficients are zero. This can change in two cases: First, hydrodynamicinteractions (see Section 4.2) lead to a more complex diffusion tensor that containsoff-diagonal terms. Second, incompressibility plays a role here. As discussed in Sec-tion 3.2.3, polymer DDFT deals, in its simplest form, with incompressible mixtures.For a binary mixture, an incompressibility constraint leads from Eq. (53), which is astandard DDFT equation for mixtures, to Eqs. (54) and (55), which, if written in theform (216), would correspond to a mobility matrix with off-diagonal elements. Thisprocedure was discussed by de Gennes [545]. Mobility matrices with off-diagonal ele-ments also arise in the thin-film hydrodynamics of incompressible binary mixtures, ina form that is similar to the one arising in incompressible polymer DDFT. Xu et al.[546] compared this form to the standard form (124) of DDFT for mixtures withoutoff-diagonal contributions and attributed this difference to the different choice of theframe of reference. While a detailed investigation of the relation between incompress-ible thin-film hydrodynamics and incompressible polymer theory is a task for futurework, the similarity of the arguments employed in the derivations and of the form ofthe resulting theories makes it likely that they are connected.

We now return to Eq. (124). The simplest case is that of a binary mixture. Asdemonstrated in Refs. [272,303,353], this allows to model phase separation. A furtherextension is to give the particle species additional degrees of freedom, such as a mag-netic moment [448–450]. Mixtures of microswimmers are considered in Ref. [411], thisallows to study the phase behavior of, e.g., pusher-puller mixtures or mixtures of activeand passive particles. Chauviere et al. [46] derived a DDFT for a system of multiplecell species and Al-Saedi et al. [49] modeled the competition of healthy and cancer cellsusing a two-species DDFT (see Section 8.1.11). The particle-conserving dynamics (seeSection 4.9) of binary mixtures is considered in Ref. [39]. One can also apply DDFTfor mixtures to one-component systems whose components can be in different states[418]. A natural application of DDFT for mixtures is the calculation of the van Hovefunction [27,51,159,203,204,293,342,343,358] (see Section 8.3).

Moreover, DDFT for binary mixtures has been applied to model anisotropic paircorrelations [286], bubble dynamics [451], chemical reactions [52,416,417,452], CO2

dissolving in poly (lactic acid) melt [453], colloid-polymer mixtures [203], demixingon a sphere [454,455], derivation of a PFC model for binary mixtures [380,456], dis-ease spreading [50,420], evaporating nanoparticle suspensions [457–460], field-drivendemixing [306], instabilities in driven systems [324,325], particles switching betweentwo states [461], pattern formation [462], polymer-particle suspensions [210], proteinadsorption [463,464], sheared polymer nanocomposites [51], sedimentation [465], se-lectivity effects [201], solvation dynamics [466–469], stratification [188,339], surfactantmixtures [419], and the Brazil nut effect [294],

47

A more general case is a multicomponent suspension, as discussed by Wittkowskiet al. [40] and Goddard et al. [317]. Both articles include a treatment of hydrodynamicinteractions. As an additional order parameter field, they use the energy density [40]and the velocity field for each particle species [317], respectively. The momentumdensity of multicomponent mixtures is also used as a dynamical variable in Refs.[385,470,471]. Grand-canonical artefacts and canonical corrections in multicomponentDDFT are discussed in Ref. [285]. A PFT for multicomponent mixtures was derived byBrader and Schmidt [27] and applied in Ref. [332]. Polymer DDFT (see Section 3.2.3)is typically set up as a multicomponent theory.

An application of DDFT for mixtures is the description of cell membranes[472,473]. Moreover, Diaw and Murillo [328] modeled an N -component astrophysi-cal plasma using DDFT. Mixtures of various protein species are considered in Refs.[333,356,474]. Finally, electrochemical systems containing ions of different charge[53,184,185,192,349,354,414,415,475–485] are an important application of DDFT formixtures (see Section 8.1.10). An option for future work is the investigation of sys-tems with nonreciprocal interactions [547].

4.6. Orientational degrees of freedom

Up to now, we have focused on passive spherical particles that (in the overdampedcase) are fully described by their center-of-mass positions. However, real particles oftenhave additional (e.g., orientational) degrees of freedom, which can be incorporated inextensions of DDFT. Orientation can be taken into account by using an extendedconfiguration space, i.e., using a density %(~r, ~u, t) that also depends on the orientation~u, or by additional order parameter fields such as a polarization field. Sources oforientational degrees of freedom are geometrical anisotropy (Section 4.6.1), magneticmoments (Section 4.6.2), and self-propulsion (Section 4.6.3).

4.6.1. Nonspherical particles

The standard DDFT formalism describes the dynamics of particles with (effective)spherical symmetry, since it is assumed in the microscopic derivations that the stateof a Brownian particle is (in the overdamped limit) characterized by its position.However, standard DDFT can also be extended to nonspherical particles. For thispurpose, we additionally need to specify the orientation of a particle. If the particlehas an axis of continuous rotational symmetry (“uniaxial particle”), this can be doneby a vector ~u which points in the direction of this axis, or, for a spherical activeparticle, in the direction of self-propulsion. The orientation vector ~u is parametrizedby one angle ϕ in 2D and by two angles ϕ and ϑ in 3D [548], which correspondsto the use of polar coordinates and spherical coordinates, respectively. If the particlehas a more general shape, its orientation is fixed by the rotation that transformsfrom the laboratory-fixed to the body-fixed frame of reference. This rotation can beparametrized in 2D by one angle ϕ [548] and in 3D by the three Euler angles ϕ, ϑ,and ς [548]. In the convention employed in Refs. [21,548–550], which is equivalent tothe second convention of Ref. [551], the first two Euler angles coincide with the anglesϕ and ϑ of the spherical coordinates. Theoretical descriptions of anisotropic particlesare reviewed in Ref. [486].

Orientational degrees of freedom were considered already at early stages of the devel-opment of DDFT. A Smoluchowski equation for orientational dynamics was obtainedby Calef and Wolynes [107], whose work was extended by Chandra and Bagchi [110]. In

48

the study of solvation [111], diffusion [487], and orientational relaxation [109,112,488],the DDFT equation

∂

∂t%(~r, ~u, t) = D~∇~r ·

(%(~r, ~u, t)~∇~r

δF

δ%(~r, ~u, t)

)+DR

~∇~u ·(%(~r, ~u, t)~∇~u

δF

δ%(~r, ~u, t)

)(125)

for the one-body density % that measures the probability of finding a particle withposition ~r and orientation ~u at time t was used. Here, D and DR are the translationaland rotational diffusion coefficients of a free particle, respectively, and ~∇~u denotes therotation operator. We here adapt the notation of Refs. [121,550] and use ~∇~u for the

rotation operator. An alternative, employed, e.g., in Ref. [43], is to use ~∇~u for a partial

derivative with respect to the elements of the orientation vector ~u, in analogy to ~∇~r. Inthis case, the rotation operator is given by ~u× ~∇~u. A review on orientational relaxationdynamics is given in Ref. [121] and a review on early forms of DDFT (including themethod from Chandra and Bagchi) can be found in Ref. [123]. The DDFT (125)can also be extended to mixtures [466]. Applications of this form of DDFT includethe calculation of orientational relaxation times [552,553], mixed quantum-classicaldynamics [88] (see Section 4.11), and protein solvation [334].

From a microscopic point of view, uniaxial particles were considered by Rex et al.[43] (see also Refs. [153,214,215]). In a DFT (or DDFT) for uniaxial particles, theorientation is incorporated by using a density field %(~r, ~u) rather than simply ρ(~r), suchthat %(~r, ~u) measures the probability of finding a particle at position ~r with orientation~u [554,555]. The derivation of a dynamic equation for %(~r, ~u) then proceeds along thesame lines as the derivation from the Smoluchowski equation outlined in Section 3.3.3.For nonspherical particles, the Smoluchowski equation reads

∂

∂tΨ(~rk, ~uk, t) =

N∑i=1

(~∇~ri ·

(D(~ui)

(~∇~ri +

1

kBT~∇~riU(~rk, ~uk, t)

))+DR

~∇~ui·(~∇~ui

+1

kBT~∇~ui

U(~rk, ~uk, t)))

Ψ(~rk, ~uk, t)).

(126)

Compared to Eq. (79), the microscopic density Ψ now also depends on the orientations~uk of the individual particles, the translational diffusion constant is now a tensor

D(~u) = D‖~u⊗ ~u+D⊥(1− ~u⊗ ~u) (127)

with parallel and perpendicular translational diffusion coefficients D‖ and D⊥, respec-tively, and identity matrix 1, and we have added a term for the rotation with thecoefficient of rotational diffusion DR. The generalized force balance

kBT

%(~r, ~u)~∇~r%(~r, ~u) + ~∇~rU1(~r, ~u) = −~∇~r

δFexc[%]

δ%(~r, ~u)(128)

of thermodynamic equilibrium used analogously in the standard derivations is nowsupplemented by a generalized torque balance

kBT

%(~r, ~u)~∇~u%(~r, ~u) + ~∇~uU1(~r, ~u) = −~∇~u

δFexc[%]

δ%(~r, ~u). (129)

49

Making an adiabatic approximation in the usual way, this gives the DDFT for uniaxialparticles

∂

∂t%(~r, ~u, t) = ~∇~r ·

(D(~u)%(~r, ~u, t)~∇~r

δF [%]

δ%(~r, ~u, t)

)+DR

~∇~u ·(%(~r, ~u, t)~∇~u

δF [%]

δ%(~r, ~u, t)

),

(130)

where the free energy F still has the form (10) (the ideal contribution is now that ofan ideal rotor gas).

For a further analysis of the field %(~r, ~u, t), it is typically helpful to perform anexpansion of the orientational dependence in symmetric traceless tensors. The generalexpansion for a function f(~u) in 3D reads [548]

f(~u) =

∞∑l=0

3∑i1,...,il=1

f(3D)i1···ilui1 · · ·uil (131)

with the coefficients

f(3D)i1···il = A

(3D)l

∫S2

dΩ f(~u)T(3D)i1···il , (132)

the normalization A(3D)l = (2l + 1)/(4π), the integration over the unit sphere

∫S2

dΩ,and the tensor polynomials

T(3D)i1···il =

(−1)l

l!∂i1 · · · ∂il

1

r

∣∣∣∣~r=~u

, (133)

where ∂i denotes the spatial derivative with respect to the i-th element of ~r. Notethat different expansions have to be used in 2D or for asymmetric particles whose one-body distribution depends on three angles. (A detailed overview over various types oforientational expansions is given in Ref. [548].) If we perform the expansion (131) forthe function %(~r, ~u, t) up to second order, we find

%(~r, ~u, t) =1

4πρ(~r, t) +

3∑i=1

Pi(~r, t)ui +

3∑i,j=1

Qij(~r, t)uiuj (134)

with

ρ(~r, t) =

∫S2

dΩ %(~r, ~u, t), (135)

Pi(~r, t) =3

4π

∫S2

dΩ %(~r, ~u, t)ui, (136)

Qij(~r, t) =15

8π

∫S2

dΩ %(~r, ~u, t)(uiuj −

1

3δij

), (137)

where ~P (~r, t) is the local polarization and Q(~r, t) the local nematic tensor. By insert-ing this expansion into Eq. (130) and converting the functional derivative with respect

50

to %(~r, ~u, t) into derivatives with respect to the fields ρ(~r, t), ~P (~r, t), and Q(~r, t), oneobtains a set of coupled dynamic equations for these fields. A detailed calculation fol-lowing this procedure can be found in Ref. [556]. (The orientational expansion methodis not only used in DDFT, but also in other soft and active matter theories [557–559].) Alternatively, one could directly derive the equations of motion for the orderparameter fields from the projection operator formalism.

The DDFT for nonspherical particles can be generalized further in two ways. First,we have assumed that the particles have an axis of continuous rotational symmetry.This includes apolar and polar particles, which both can be described using an orien-tation vector ~u (apolarity is reflected by a symmetry %(~u) = %(−~u) [548]). The moregeneral case would be a particle with arbitrary shape, whose orientation, as mentionedabove, is described by a rotation parametrized by the three Euler angles ϕ, ϑ, andς. For convenience, we summarize the Euler angles in a vector ~$ = (ϕ, ϑ, ς)T. Sec-ond, we can consider active particles, as done, e.g., by Wensink and Lowen [20] foractive rods. These have, due to their self-propulsion, always an intrinsic orientation.Self-propulsion is an additional extension, since it leads to a change of the dynami-cal structure of DDFT (see Section 4.6.3). Both cases were considered by Wittkowskiand Lowen [21], whose result is discussed in Section 4.6.3. A theory involving hydrody-namic interactions, inertia, and particles of general shape was derived using projectionoperator methods by Duran-Olivencia et al. [401].

Bier and van Roij [183, 489] studied platelike colloids in a phenomenological model.Self-diffusion in smectic and nematic liquid crystals is studied in Refs. [358,359]. Non-Gaussian diffusion in such systems is observed in Ref. [358] as a result of temporarycages. Similar behavior is also found in simulations of equilibrium smectic phases ofhard rods [560,561] and relaxation of spherocylinders [562,563].

The application of FMT (see Section 2.2.2) to hard spherocylinders in DDFT isdiscussed in Refs. [175,176]. Uematsu and Yoshimori [490] studied polarization re-laxation using a DDFT with orientational dynamics. In Ref. [239], Dean’s stochasticDDFT equation (50) is extended to dipolar particles. Duran-Olivencia et al. [383] de-rived a general fluctuating DDFT for particles with arbitrary shape that also includesmomentum and angular momentum density as relevant variables. The growth of hard-rod monolayers is analyzed in Ref. [181], including a consideration of nematic order.In Ref. [491], the nucleation of hard squares is studied.

DDFTs for nonspherical particles can also be used to obtain PFC equations thatinclude orientational dynamics [492,493] or anisotropic particle interaction potentials[494] (see Section 6.3). Finally, the methods presented in this section are also relevantfor magnetic particles [448,450] (see Section 4.6.2).

4.6.2. Magnetic particles

In order to describe continuum systems with magnetic order, DDFT has also beenextended to systems of particles with a (classical) spin ~σs. These theories have beenused to study phase separation in fluids containing magnetic particles [448–450]. Thespin, which is another degree of freedom of the particles in addition to the position,is normalized and described by an orientation specified by the spherical coordinateangles ϑ and ϕ or an orientation vector ~u. Thus, the description of magnetic particlesin DDFT is completely analogous to the description of nonspherical particles discussedin Section 4.6.1. For the spin contribution to the interaction potential, one frequently

51

uses the Heisenberg model

U2(‖~r − ~r′‖, ~u, ~u′) = J (‖~r − ~r′‖)~σs · ~σ′s, (138)

where the sign of the coupling J determines whether ferromagnetic or antiferromag-netic ordering is preferred.

The stochastic dynamics of magnetic dipoles is considered in Ref. [239]. A DFTfor ferrogels (magnetic colloids in a polymer matrix) was derived in Refs. [564,565].Moreover, the DFT of ferrofluids is discussed in Refs. [566–571]. Methods of thistype provide a possible starting point for future work on applications of DDFT tomagnetism.

4.6.3. Active particles

DDFT can also be applied to study systems of active particles. Field theories for ac-tive matter can, in general, be classified into dry models, which model active particleswithout explicitly considering the solvent, and wet models, where hydrodynamics isalso considered. General reviews on active matter can be found in Refs. [252,486,572–582]. A particular challenge in DDFT is that active particles are far-from-equilibriumsystems whose dynamics cannot be assumed to be driven by the gradient of the equi-librium free energy. Possible solutions are to add to the standard DDFT for passivesystems (with passive free energy) an active term or to derive an “effective” free energythat describes (a steady state of) the active system. A DDFT model for dry activeparticles of the former type was obtained for uniaxial particles by Wensink and Lowen[20] and for general shapes by Wittkowski and Lowen [21]. These models only hold forthe case of weak activity.

We start by discussing the DDFT for active particles with general shape by Wit-tkowski and Lowen [21]. The density is now a function %(~r, ~$, t) and the Smoluchowskiequation reads

∂

∂tΨ(~rk, ~$k, t) =

N∑i=1

(~∇~ri~∇~$i

)·((

DTTi (~$i) DTR

i (~$i)DRTi (~$i) DRR

i (~$i)

)(

1

kBT

(~∇~ri~∇~$i

)U(~rk, ~$k, t)−

1

kBT

(~FA,i(~ri, ~$i, t)~TA,i(~ri, ~$i, t)

)

+

(~∇~ri~∇~$i

)))Ψ(~rk, ~$k, t)

(139)

with the angular gradient operator for particles with arbitrary shape

~∇~$ =

− cos(ϕ) cot(ϑ)∂ϕ − sin(ϕ)∂ϑ + cos(ϕ) csc(ϑ)∂ς− sin(ϕ) cot(ϑ)∂ϕ + cos(ϕ)∂ϑ + sin(ϕ) csc(ϑ)∂ς

∂ϕ

(140)

and the derivative operators ∂ϕ = ∂/∂ϕ, ∂ϑ = ∂/∂ϑ, and ∂ς = ∂/∂ς. In theSmoluchowski equation, we have used a very general diffusion matrix with elementsDTTi (~$i), D

TRi (~$i), D

RTi (~$i), and DRR

i (~$i) that involves couplings between transla-tion and rotation, in addition to the purely translational and rotational diffusion,but no hydrodynamic interactions. The precise form of the elements depends on

52

the particle shape. (Various shapes are discussed in Ref. [583], see Ref. [584] forthe accompanying software.) Furthermore, the Smoluchowski equation contains the

active force ~FA,i(~ri, ~$i, t) = R−1(~$i)~FA,0,i(~ri, t) and active torque ~TA,i(~ri, ~$i, t) =

R−1(~$i)~TA,0,i(~ri, t) with the rotation matrix R(~$), the propulsion force ~FA,0,i(~ri, t),

and the propulsion torque ~TA,0,i(~ri, t), where the latter two can be space- and time-dependent. After integrating over positional and orientational degrees of freedom ofall particles except for the first one and performing an adiabatic approximation, onearrives at the generalized DDFT equation

∂

∂t%(~r, ~$, t) =

1

kBT

(~∇~r~∇~$

)·((

DTT(~$) DTR(~$)DRT(~$) DRR(~$)

)(%(~r, ~$, t)

((~∇~r~∇~$

)δF [ρ]

δ%(~r, ~$, t)−(~FA(~r, ~$, t)~TA(~r, ~$, t)

)))),

(141)

where we have omitted the index 1 for the quantities ~r, ~$, DTT(~$), DTR(~$), DRT(~$),

DRR(~$), ~FA, and ~TA. (In principle, the elements DTT(~$), DTR(~$), DRT(~$), andDRR(~$) of the short-time diffusion tensor could also be space-dependent [585].)

Menzel et al. [22] derived a DDFT for hydrodynamically interacting microswimmers.Here, the individual swimmers are modeled as spherical particles that exert a force onthe surrounding fluid at two force centers located along an orientation vector ~u. Thisleads to self-propulsion. Assuming low Reynolds number, the fluid flow velocity ~v(~r, t)is governed by Stokes’ equation

− ηs~∇2~v(~r, t) + ~∇p(~r, t) =

N∑i=1

~fi(~ri, ~ui, t) (142)

with the shear viscosity ηs, pressure p(~r, t), and force density ~fi(~ri, ~ui, t) exerted bythe i-th swimmer. Since the swimmers influence the fluid flow by the force they apply,but at the same time are dragged along by the fluid, they self-propel and interacthydrodynamically.

The derivation of the DDFT then proceeds in a similar way as in the treatmentof nonspherical particles discussed in Section 4.6.1, since the one-body density foractive particles also needs to take into account the orientation. With the translationalvelocity ~vi and angular velocity ~ωi of the i-th particle resulting from the flow field, thecontinuity equation for the N -body density Ψ(~rk, ~uk, t) is given by

∂

∂tΨ(~rk, ~uk, t) = −

N∑i=1

~∇~ri ·(~viΨ(~rk, ~uk, t))+ ~∇~ui·(~ωiΨ(~rk, ~uk, t)). (143)

The hydrodynamics is contained in the expressions for the velocities ~vi and ~ωi, whichcan be expressed in terms of the hydrodynamic force and torque, respectively. Byintegrating over the coordinates of N − 1 particles and applying the usual adiabaticapproximation, one can then derive a DDFT for the microswimmer system. (Theexact equations are rather lengthy and can be found in Ref. [22].) This approach hasalso been extended towards circle swimmers [409], polar ordering [410], and multi-species microswimmer suspensions [411]. It also holds, being based on an equilibriumapproximation (as standard DDFT), only for weak activity.

53

Another DDFT for active particles was obtained, based on effective free energies foractive steady states rather than passive free energies, by Enculescu and Stark [348].Wittmann and Brader [23] constructed a DDFT-based effective equilibrium descriptionthat predicts certain phase transitions of active Brownian particles (ABPs). It is basedon the work in Refs. [298,495] and reads

∂

∂tρ(~r, t) =

D

kBT~∇ ·(ρ(~r, t)~∇δFeff [ρ]

δρ(~r, t)

)(144)

with the translational diffusion coefficientD. Note that this type of active DDFT differsin an interesting way from the type for which Eq. (141) is a paradigmatic example.Equation (141) provides a dynamic equation for %(~r, ~$, t) rather than ρ(~r, t), i.e.,the orientational degrees of freedom are considered explicitly, and it incorporates theactivity using an additional term in the final DDFT equation. In contrast, Eq. (144)looks exactly like the standard DDFT equation (35). The activity is incorporatedin Eq. (144) by the fact that an effective free energy Feff is used. It is obtained byreplacing the actual external potential by an effective external potential that consistsof the actual external potential U1 and a contribution that results from integrating anapproximate expression for the steady-state active force. Effective equilibrium statesare also considered in Refs. [205,496]. A DDFT for active particles driven by colorednoise was derived by Wittmann et al. [205].

Pototsky and Stark [298] derived a DDFT for active particles by mapping themonto a passive system with a modified potential. Menzel et al. [493] obtained an activePFC model from an active DDFT (see Section 6.3). Sharma and Brader [297] havedeveloped a DDFT for ABPs with spatially inhomogeneous activity (building up onwork in Ref. [497]). Paliwal et al. [299] constructed a chemical-potential-like functionfor ABPs in a steady state to obtain a DDFT-like model. Activity-driven spins weremodeled by Zakine et al. [461]. Krinninger and Schmidt [296] derived a PFT for ABPs(see Section 5.2.2). Arold and Schmiedeberg [498] proposed a DDFT for active particleswith inertia that includes the polarization as a dynamical variable. A reaction DDFT(see Section 8.2.5) was used by Moncho-Jorda and Dzubiella [418] to model activeswitching.

4.7. Momentum

A further microscopic degree of freedom is the momentum, which is of importance if theparticles are not overdamped. Just like the description of particles with orientationaldegrees of freedom requires a density %(~r, ~u, t) rather than ρ(~r, t) in the microscopicdescription, momentum degrees of freedom require an extension to a phase-space dis-tribution function Ψ(~r,~v, t) depending on the velocity ~v of the particles. Moreover, asin the case of orientation, an expansion of this extended density is possible that leadsto an infinite hierarchy of order parameter fields. At lowest orders, this gives rise tothe one-body density ρ(~r, t) and the momentum density ~g(~r, t).

4.7.1. Inertia

The effects of inertia in the DDFT of (one-dimensional) colloidal fluids were studiedby Marini Bettolo Marconi and Tarazona [42]. The analysis is based on the phase-space distribution function Ψ(~r,~v, t), whose dynamics follows the (1D) Fokker-Planck

54

equation

∂

∂tΨ(x, v, t) +

(v∂

∂x+F

m

∂

∂v

)Ψ(x, v, t) = γ

(∂

∂vv +

T

m

∂2

∂v2

)Ψ(x, v, t) + C(x, v, t,Ψ2),

(145)

where C is a collision operator depending on the two-body distribution Ψ2 that canbe approximated by the assumption of molecular chaos. The dynamic equation is thenrewritten in terms of nondimensionalized parameters. This allows for a time-scaleseparation in terms of a dimensionless parameter γnd (proportional to γ) measuringthe relaxation (see also Section 7.2.2). At lowest nontrivial order, one recovers the usualoverdamped DDFT. These results were generalized to arbitrarily many dimensions inRef. [499]. The described method for incorporating inertia into DDFT can be used tosystematically obtain corrections to the overdamped case [500].

A DDFT for atomic fluids, where, unlike in colloidal fluids, inertia always playsa role, has been derived by Archer [131] (general fluids were also treated by Chanand Finken [130], see Section 5.4.2). Starting from Newton’s equation of motion andperforming an ensemble average, one finds for the one-body density ρ(~r, t) the exactequation of motion

∂2

∂t2ρ(~r, t) +B(~r, t) =

kBT

m~∇2ρ(~r, t) +

1

m~∇ ·∫

d3r′ 〈ρ(~r, t)ρ(~r′, t)〉 ~∇U2(~r − ~r′)

+1

m~∇ · (ρ(~r, t)~∇U1(~r, t)) (146)

with the term

B(~r, t) =1

(2π)3

∫d3k k2

⟨ N∑i=1

αi(t)ei~k·~ri⟩e−i~k·~r, (147)

where m is the mass of the particles, ~ri is the position of the i-th particle, and αi(t) is

the deviation of (~ri(t) ·~k)2/k2 with the wavevector ~k from its equilibrium mean value.Equation (146) is closed with two approximations: First, the two-particle correlationsare, as in standard DDFT, replaced by their equilibrium values, which are related tothe functional derivative of the equilibrium free energy. Second, the term B(~r, t), whichvanishes in equilibrium, is assumed to have the form νρ(~r, t) with an undeterminedfrequency ν. This gives the result

∂2

∂t2ρ(~r, t) + ν

∂

∂tρ(~r, t) =

1

m~∇ ·(ρ(~r, t)~∇ δF [ρ]

δρ(~r, t)

). (148)

DDFTs with inertia can also be shown to possess a H-theorem [393]. They can beextended towards hydrodynamic interactions [317,319,320,400,401] and orientationaldegrees of freedom [401]. Moreover, a DDFT with inertia has been developed for activeparticles [498]. One can also incorporate inertia in PFT [29] (see Section 5.2.3). Ap-plications of inertial DDFT include the study of sound waves [131] (see Section 8.2.8),mode coupling theory [131] (see Section 6.1), glassy behavior [501], granular systems[341,500], freezing [502], and dielectricity [503].

55

4.7.2. Momentum density

An important extension of DDFT are more general theories that describe not only thedensity ρ(~r, t), but also the momentum density ~g(~r, t). These theories typically relateρ(~r, t) to the flux of ~g(~r, t) by the continuity equation

∂

∂tρ(~r, t) +

1

m~∇ · ~g(~r, t) = 0. (149)

Furthermore, they give a separate equation governing the momentum density, whichhas the form of a (generalization of the) Navier-Stokes equation [88]

∂

∂t~g(~r, t) = −ρ(~r, t)~∇ δF [ρ]

δρ(~r, t)+ ~D[~g] (150)

with a dissipative contribution ~D. These theories then contain standard DDFT as alimiting case and can thus be seen as generalizations of DDFT towards hydrodynamics.(Historically, however, they also formed a basis for the derivation of overdamped DDFT[11], see Section 3.3.1.) Eliminating the velocity or momentum density field from thecoupled equations gives a second-order equation of motion for the one-body density[402,504].

Frequently, such theories are derived from kinetic theories that model the phase-space distribution Ψ(~r,~v, t) rather than just the one-particle density ρ(~r, t) [88,143,385,471,477,505,506,508]. These results contain DDFT as the high-friction limit. Suchmethods allow to relate the well-known continuum theories of fluid mechanics tothe microscopic particle dynamics, which is important, e.g., for modeling flow in mi-crochannels [507]. Moreover, they can be seen as arising from a truncated expansion ofthe full phase-space dynamics in moments of the momentum (that can also be stoppedat higher orders, in this case the kinetic energy density tensor, which contains the ki-netic pressure tensor, is also among the dynamical variables) [508]. Theories includingthe momentum density can also be derived using the projection operator formalism byprojecting the microscopic dynamics onto mass and momentum densities as relevantvariables. This is done for simple fluids by Camargo et al. [384]. A further explorationof this approach can be found in Refs. [428,509,534].

One can further generalize DDFT with momentum density, e.g., by using additionalorder parameter fields. Dynamic equations that include, in addition to mass and mo-mentum density, also the energy density, are derived in Refs. [186,385,505,510,522].Majaniemi and Grant [511] coupled mass and momentum density to a dynamic equa-tion for a displacement field. Mixtures are considered in Refs. [317,385,451]. One canalso derive the coupled equations for mass and momentum density from quantum me-chanics [88,512,513]. Diaw and Murillo [47] applied DDFT to quantum hydrodynamics(see Section 4.11).

Archer [504] derived a theory for mass density and flow field that contains as limitingcases both overdamped DDFT and the Euler equation. Rotenberg et al. [478] coupleda DDFT-like equation for charged systems to the Navier-Stokes equation. A couplingto the Navier-Stokes equation is also possible for the PNP equations [414,415] (seeSection 8.1.10). Another theory combining mass and momentum density is the “kineticdensity functional theory” derived in Ref. [502], which is used to model the liquid-solidtransition. The momentum density is also included in DDFT-based theories of plasmadynamics [327,328] (see Section 8.1.7). Praetorius and Voigt [514] derived, starting

56

from DDFT, a PFC model coupled to a dynamic equation for the flow field. A PFCequation coupled with a Navier-Stokes-Cahn-Hilliard equation is derived in Ref. [515](see also Ref. [516]). Arold and Schmiedeberg [498] included the momentum densityin an active matter model. The momentum density is considered in the context ofstochastic theories in Refs. [257–259,517,518]. Numerical methods for equations ofthis type can be found in Ref. [519].

4.7.3. Kinetic theory

For the description of inertia and momentum transport, it is important to include notonly the positions, but also the momenta or velocities of the particles in a dynamicaltheory. Hence, a natural extension of DDFT is to consider the phase-space distributionΨ(~r,~v, t) rather than the spatial distribution ρ(~r, t). A first theory of this type, whichis only applicable to dilute gases, has already been developed by Boltzmann [89]. Morerecently and more generally, a dynamic theory for Ψ(~r,~v, t) has been derived using theprojection operator formalism by Anero and Espanol [392]. Perez-Madrid et al. [259]have obtained a Fokker-Planck equation for the phase-space distribution functional.A review of kinetic theory can be found in Ref. [477].

Starting from a phase-space theory, it is possible to derive DDFT as an overdampedlimit, based on the assumption that the velocity very quickly reaches its steady state[143]. Phase-space methods also give inertial corrections to DDFT [291,499]. Therelation to overdamped DDFT can be obtained using a multiple-time-scale analy-sis [341,500] (see Section 7.2.2). A particularly important application of kinetic the-ory is the microscopic description of fluid dynamics [477]. Kinetic theory also al-lows to obtain a “kinetic density functional theory” for the description of the solid-liquid phase transition [502]. Moreover, a phase-space description connects DDFT tothe Boltzmann theory [505]. From the phase-space description, one can derive theequations of hydrodynamics [506]. This is also possible for multicomponent systems[385,470,471,477,481,482] and granular media [186,341].

4.8. Nonisothermal systems

While standard DDFT assumes the system to have a fixed temperature, extensionsto nonisothermal systems have also been derived. This topic has a clear connection tothe case of momentum discussed in Section 4.7, since a natural dynamical variable fornonisothermal systems is the energy density. In many hydrodynamic theories, it is usedas an order parameter field in addition to mass and momentum density, which allowsto describe nonisothermal systems. The reason why the energy density is importantis that energy – like mass and momentum – is a conserved quantity, such that theenergy density can often be assumed to be a slow variable. As can be shown usingprojection operator methods [279] (see Section 5.3.1), this implies that the energydensity is important for the dynamics of the system.

4.8.1. Fixed temperature gradients

Treatments of nonisothermal systems can either consider externally fixed tempera-ture gradients (this is the simpler case) or couple DDFT to an additional dynamicequation for the local temperature. Lopez and Marini Bettolo Marconi [134] deriveda DDFT-like transport equation for particles in a nonuniform heat bath with temper-

57

ature distribution T (x) (in 1D), which reads

∂

∂tρ(x, t) = ~∇ ·

(~∇(D(x)ρ(x, t)) +

1

γρ(x, t)~∇

(δFexc

δρ(x, t)+ U1(x)

))(151)

with D(x) = kBT (x)/γ. The derivation is sketched in Section 7.2.2. This result is alsogiven by Marini Bettolo Marconi et al. [500].

4.8.2. Energy density

A description in terms of a single order parameter field (such as the one-body densityin DDFT) implies that all other degrees of freedom (such as the local temperature)relax very quickly. Hence, in order to extend DDFT to nonisothermal systems, onerequires an additional order parameter field. Natural candidates are the local energydensity [40,138] and the local entropy density [520].

Microscopic derivations of DDFTs for nonisothermal systems can be performed inthe projection operator formalism (see Section 5.3.1) by projecting onto the energydensity as a second relevant variable in addition to the number density. This is theidea behind EDDFT [40,212] and FTD [138], both of which are extensions of DDFTtowards nonisothermal systems. They are discussed in Paragraphs 5.3.2.1 and 5.3.2.2,respectively. Moreover, an extended theory involving mass, momentum, and energydensity as well as a dynamic equation for the two-body correlation was derived byZhao and Wu [510] (see also Ref. [521]). The dynamics of the energy density is alsoconsidered in Refs. [385,505,522]. Coupled equations for heat and mass transport (thelatter formally identical to DDFT) were obtained by Kocher and Provatas [586] fromnonequilibrium thermodynamics. The temperature of granular fluids was studied byGoddard et al. [186].

4.8.3. Entropy density

The entropy density was considered by Schmidt [520] based on a phenomenologicalgeneralization of an equilibrium formalism: An energy functional E[ρ, s] is defined thatdepends not only on the number density ρ(~r) but also on the entropy density s(~r). Itsatisfies the equilibrium Euler-Lagrange equations

δE[ρ, s]

δρ(~r)

∣∣∣∣ρeq,seq

= µ− U1(~r), (152)

δE[ρ, s]

δs(~r)

∣∣∣∣ρeq,seq

= T (153)

with the equilibrium number density ρeq(~r), equilibrium entropy density seq(~r), chem-ical potential µ, and temperature T . Assuming the Gibbs-Duhem relation (solved fords)

ds =1

Tde− µ

Tdρ (154)

58

to be valid, one can calculate the rate of change of the entropy as

∂

∂ts(~r, t) =

1

T (~r, t)

∂

∂te(~r, t)− µ(~r, t)

T (~r, t)

∂

∂tρ(~r, t), (155)

where e is the energy density. One then identifies temperature T and chemical po-tential µ as the driving forces for changes of energy density e and number densityρ, respectively. Inserting these driving forces into a general continuity equation andexpressing temperature and chemical potential as functional derivatives of E with re-spect to entropy and number density, respectively, then gives a dynamic equation forthe entropy density.

A similar approach is used by Wittkowski et al. [212] based on a microscopic deriva-tion in EDDFT. It is based on the framework of linear irreversible thermodynamicsdiscussed in Section 6.2. Here, Eq. (154) is written for a general set of conserved andnonconserved variables ai rather than just for ρ(~r, t). This leads to a generalizationof Eq. (155) given by

∂

∂ts(~r, t) =

1

T (~r, t)

∂

∂te(~r, t)−

κ∑i=1

1

T (~r, t)

δF [ai]δai(~r, t)

∂

∂tai(~r, t). (156)

The projection operator formalism allows to obtain the transport equations for thesevariables in terms of their thermodynamic conjugates. Since the entropy is a noncon-served variable, its dynamics can be expressed in the general form

∂

∂ts(~r, t) = ~∇ · ~Js(~r, t)− Φs(~r, t) (157)

with the entropy current ~Js(~r, t) and the entropy production

Φs(~r, t) =2r(~r, t)

T (~r, t), (158)

where r(~r, t) is the dissipation function (see Section 6.2). Note that all these approachesare based on the local validity of the laws of thermodynamics, which do not hold forsystems far from equilibrium.

Finally, Hutter and Brader [522] employed (in nonequilibrium thermodynamics) anentropy density s(~r, t) that is a functional of mass and energy density and obtained adynamic equation for s. Entropy production has also been discussed in the context ofPFC models [587].

4.9. Particle-conserving dynamics

We now turn to a different type of extension, which does not aim to incorporateadditional order parameter fields. Instead, it corrects an inaccuracy arising from one ofthe approximations involved in DDFT (see Section 3.4.1): Although the canonical andthe grand-canonical ensemble are equivalent in the thermodynamic limit, inaccuraciescan arise when treating systems with a finite – and, in particular, small – fixed numberof particles in the grand-canonical ensemble [588,589]. While DFT is a grand-canonicaltheory, DDFT is not completely consistent in this regard. In principle, it is a canonical

59

theory that is based on the free energy F and conserves the particle number as it hasthe form of a continuity equation. However, in practice it uses thermodynamic forcesbased on a grand-canonical free energy. This can lead to differences between DFT andDDFT, e.g., in the prediction of nucleation pathways [448]. Agreement between DFTand DDFT in the equilibrium case is only guaranteed in the thermodynamic limit[465]. A comparison of canonical and grand-canonical results is made in Ref. [397](see also Ref. [426]) for particles confined to a cylinder, where in the canonical case aLagrange multiplier is used.

To overcome problems of the grand-canonical ensemble, the method of particle-conserving dynamics (PCD) [38] uses a canonical framework for DDFT. We startby discussing the equilibrium case. While the canonical free energy functional is notexplicitly available, one can convert between the grand-canonical partition function Ξand the canonical partition function ZN for a system with N particles using [523]

Ξ =

∞∑N=0

eβµNZN . (159)

Moreover, the grand-canonical density ρµ(~r) corresponding to the chemical potentialµ is related to the canonical density ρN (~r) for a system with N particles by

ρµ(~r) =

∞∑N=0

pN (µ)ρN (~r), (160)

where

pN (µ) =1

ΞeβµNZN (161)

is the probability of finding N particles for a given chemical potential µ. Starting fromthese relations and the grand-canonical Euler-Lagrange equation, an iterative schemecan be developed that gives the external potential Upcd(~r) creating a density ρ(~r) incanonical equilibrium. This leads to the canonical intrinsic free energy

FN [ρ] = −kBT ln(ZN )−∫

d3r ρ(~r)Upcd(~r). (162)

The dynamic equation of the canonical density ρN is given by

∂

∂tρN (~r, t) = D~∇ ·

(~∇ρN (~r, t)− 1

kBT~fN (~r, t)− 1

kBTρN (~r, t)( ~X(~r, t)− ~∇U1(~r, t))

),

(163)

where ~fN is the internal force density arising from interactions, which in DDFT isapproximated by the adiabatic force ~fad

N , and ~X is a nonconservative force acting onthe particles. Obtaining the dynamics within the DDFT approximation then requiresto find an expression for ~fad

N . Various options are discussed in Ref. [38]. One of themis

~fadN (~r, t) = −ρad

N (~r, t)~∇δFexcN [ρ]

δρ(~r, t)

∣∣∣∣ρ(~r,t)=ρadN (~r,t)

(164)

60

with the one-body density ρadN (~r, t) = ρN (~r, t) of the adiabatic reference system and

the canonical excess free energy F excN .

Extensions of PCD have also been derived. In Ref. [39], PCD was generalized tobinary mixtures. Here, it was shown that PCD does not conserve particle order forhard rods in 1D. To solve this problem, the theory of order-preserving dynamics (OPD)was developed based on PCD in Ref. [273] (see Section 8.1.5).

4.10. Extensions of DDFT for polymer melts

Extensions have also been developed for the polymer DDFT discussed in Section 3.2.3.We present these extensions here in a separate section, since, historically, this form ofDDFT has had a separate development.

A useful approximation is provided by the method of “external potential dynam-ics” (EPD), which was introduced by Maurits and Fraaije [268]. As discussed in Sec-tion 3.2.3, the dynamic equations governing polymer dynamics are typically nonlocal.An example is the Rouse dynamics, which is given by Eq. (56). In EPD, it is possibleto rewrite this nonlocal equation in a local form while keeping the nonlocal coupling,provided that a certain approximation can be made. What is exploited here is theexistence of a bijective relation between density fields ρi and external potentials Ui.The approximation is that, for the two-body correlator Pij in Eq. (56), we can write

~∇~rPij(~r, ~r′, t) = −~∇~r′Pij(~r, ~r′, t). (165)

This allows to write

~∇~r ·∫

d3r′ Pij(~r, ~r′, t)~∇~r′µj(~r′, t) =

∫d3r′ Pij(~r, ~r′, t)~∇2

~r′µj(~r′, t). (166)

The relation

kBTδρi(~r, t)

δUj(~r′, t)= −Pij(~r, ~r′, t) (167)

gives by applying the chain rule

∂

∂tρi(~r, t) =

Npt∑j=1

∫dr′

δρi(~r, t)

δUj(~r′, t)∂

∂tUj(~r

′, t)

= − 1

kBT

Npt∑j=1

∫dr′ Pij(~r, ~r′, t)

∂

∂tUj(~r

′, t).

(168)

Combining Eqs. (56), (166), and (168) leads to

∂

∂tUi(~r, t) = − Dro

kBT~∇2µi(~r, t), (169)

such that the nonlocal equation (56) has been replaced by a local equation. This allowsfor an efficient numerical solution [268,590–593]. A more exact scheme, along with acomparison to BD simulations, can be found in Ref. [322].

61

An extension towards compressible copolymer melts was developed in Ref. [126].Hydrodynamic effects were considered by Maurits et al. [524], who added a streamingterm to Eq. (53). This gives

∂

∂tρi(~r, t) =

Dlo

kBT~∇ · (ρi(~r, t)~∇µi(~r, t))− ~∇ · (ρi(~r, t)~v(~r, t)), (170)

where the velocity field ~v can be approximated using Darcy’s law. Moreover, polymerDDFT was extended to include viscoelastic effects in Ref. [525].

4.11. Quantum mechanics

As discussed in Section 5.4.1, density functional methods can also be applied to quan-tum systems in the form of TDDFT. This is an area on its own, which is beyond thescope of this review. Nevertheless, there are certain theories that relate quantum andclassical DDFT methods. Examples of such methods are presented here. A furthermethod of this type is quantum PFT, which is presented in Section 5.2.4.

One way to obtain a relation between classical and quantum many-particle systemsis the use of Wigner functions [594]. These allow for an exact reformulation of quantummechanics as a dynamical theory on classical phase space, where the Wigner functionW(x, p), which is related to the statistical operator ρs by the transformation (in 1D)

W(x, p) =1

2π~

∫dx′⟨x− 1

2x′|ρs|x+

1

2x′⟩ei x

′p~ (171)

with the reduced Planck constant ~, replaces the classical distribution function. Similartransformations exist for orientational degrees of freedom [595]. From the quantumLiouville equation, an equation of motion for the Wigner function can be derived thatis governed by a generalized Poisson bracket.

For the description of mixed quantum-classical dynamics, one performs the trans-formation (171) only for a subset of the degrees of freedom of the complete system.The result is a partially Wigner transformed equation of motion. Introducing a clas-sical limit by linearizing the Poisson bracket operator in ~ (formally justified by thesmall mass ratio between quantum and classical parts), one obtains a hybrid dynamicequation describing classical systems coupled to quantum systems that contains theclassical Liouville equation as a limiting case [88,596,597]. Further background infor-mation can be found in Refs. [598–600].

This method can be used to obtain a mixed quantum-classical (T)DDFT that in-cludes the momentum density (and the polarization field, if orientational relaxationis of interest) as dynamical variables [88]. The kinetic pressure can, in certain cases,be written as the functional derivative of an entropy functional. On this basis, a freeenergy functional can be defined that then governs the time evolution of density andcurrent. This relates these theories to a DDFT with an equation for the current [512].They can be applied, e.g., to nonpolar solvation dynamics [513].

Moreover, classical DDFT has been applied to quantum hydrodynamics [47]. Thegeneral transport equations for a dense electron plasma are closed adiabatically byassuming for the free energy the equilibrium form. All thermodynamic information isencoded in the free energy functional. The resulting theory has a close connection toBohmian quantum hydrodynamics.

62

5. Exact approaches generalizing DDFT

5.1. General aspects

Although DDFT is a highly successful theory, it is nevertheless only approximatelyvalid. Various limitations are presented in Section 3.4.1. Often, these arise from theadiabatic approximation (see Section 3.3), which is a central step in the derivation ofDDFT. As discussed Section 4, one can derive extended forms of DDFT to obtain alarger domain of applicability.

For going beyond standard DDFT, it is often helpful to use formalisms that pro-vide an exact description of the underlying many-body dynamics. While the resultingequations of motion can typically not be solved exactly, they are important for at leasttwo reasons: First, studying how DDFT can be obtained from them as a limiting casegives conceptual insights into the approximations involved in the derivation of DDFTand the corresponding limitations. Second, the exact equations of motion can serveas a starting point for the development of new approximate transport equations thatimprove on the limitations of DDFT. An overview over the literature on this topic isgiven in Table 1.

Forces that are not incorporated in deterministic DDFT are known as superadiabaticforces. (Stochastic DDFT is not necessarily based on the same type of approximationas deterministic DDFT and can therefore include, e.g., memory effects not present inthe deterministic formalism. An obvious example is Dean’s DDFT, which is formallyexact and therefore contains all superadiabatic forces.) The importance of superadia-batic forces was discussed by Fortini et al. [287], who developed a numerical schemefor their determination. It is based on calculating the exact force integral as well asthe fictitious “adiabatic” potential that generates the instantaneous nonequilibriumdensity. The difference between the exact and the adiabatic force integral is then thesuperadiabatic contribution. BD simulations show differences to DDFT results, re-sulting from neglecting the superadiabatic forces in the latter theory. The numericalscheme is applied to states with a variety of initial conditions in Ref. [340]. Problemsarising from the adiabatic approximation involve wrong estimates of relaxation timesand difficulties in the description of glass transition and shear flow [26]. The case ofshear was considered in Refs. [528,529], where superadiabatic forces were further clas-sified into viscous and structural forces. Superadiabatic forces also arise in quantummechanics [526].

A particularly important superadiabatic effect is memory. While it is typically notpresent on the level of single-particle dynamics (Langevin or Hamiltonian equations),reduced descriptions generally lead to a history dependence. This can be shown usingthe projection operator formalism [37,329]. A procedure for incorporating memory intogradient dynamics (in addition to the ones discussed in this section) was suggested byGalenko and Jou [601] and Koide et al. [602]. It is based on replacing the instantaneouscurrent of the system by a time convolution to take the finite propagation speed ofsignals into account. This allows to study the effects of memory and local nonequilib-rium on spinodal decomposition [603–606]. As discussed by Archer [131, 504], applyingthis procedure to DDFT gives a telegrapher’s equation that is identical in form to theDDFT for atomic fluids with inertia presented in Section 4.7.1. The same result isachieved when starting from a damped Euler equation [237]. A further discussion ofmemory effects can be found in Ref. [607].

Superadiabatic effects are also relevant for the calculation of correlation functions, asdiscussed in Ref. [31]. A nonequilibrium Ornstein-Zernike relation relates the supera-

63

diabatic current to the direct time-correlation function. Hence, superadiabatic forcestake memory effects into account [24,25]. In Refs. [25,27], the superadiabatic/non-Markovian dynamics of the van Hove function is discussed. Treffenstadt and Schmidt[329] describe superadiabatic forces in a Brownian liquid, where a resulting memory-induced motion reversal is observed.

Superadiabatic forces can be studied using exact theories. A theory of this type isthe stochastic DDFT by Dean [105], which has already been discussed in Section 3and which forms the starting point of the derivation of deterministic DDFT by MariniBettolo Marconi and Tarazona [12]. In this section, we present three additional exactapproaches. The first one is power functional theory (Section 5.2), which was derivedby Schmidt and Brader [26] as an exact description of Brownian dynamics and whichhas subsequently been extended to Newtonian mechanics [29] (see Section 5.2.3) andquantum mechanics [28,526] (see Section 5.2.4). It is based on a variational principlethat is formulated for the current rather than for the density. DDFT can be obtained bysetting the excess power dissipation (see Section 5.2.1) to zero. Consequently, studyingthe excess power dissipation allows to gain insights into superadiabatic forces.

The second one is the projection operator formalism (Section 5.3), developed byNakajima [33], Zwanzig [35], and Mori [34]. It allows to derive exact equations ofmotion for the density – and for any other dynamical variable or set of variables –by projecting the full microscopic dynamics onto the subset of variables that is ofinterest. These equations do, in general, contain a memory term. In the derivation ofDDFT using this method (see Section 3.3.4), memory effects are dropped (Markovianapproximation). The projection operator method therefore allows to extend DDFTby enlarging the set of relevant variables or by keeping memory effects. Moreover,it is very important for understanding the relation of stochastic and deterministicforms of DDFT. Applications of the projection operator formalism range far beyondsoft matter physics and include fields such as solid state theory [608], spin relaxationtheory [36,609,610], and particle physics [611,612].

A third approach starts from the Runge-Gross theorem (Section 5.4). This result,derived by Runge and Gross [81], is the basis of quantum time-dependent densityfunctional theory (TDDFT), a formally exact extension of quantum DFT towardstime-dependent systems. Chan and Finken [130] derived a similar result for classicalsystems. Since it is nonconstructive, this result has not led to significant further de-velopments in DDFT. Nevertheless, it is often considered an important result as anexistence proof [19,201,288,317].

5.2. Power functional theory

5.2.1. Standard power functional theory

Power functional theory (PFT) is an important exact generalization of DDFT. Unlike

standard DDFT, it is based on the one-body current ~J(~r, t). A power functional that

depends on both ρ(~r, t) and ~J(~r, t) is introduced. The equations of motion are thenobtained by a variation of the power functional.

We start by outlining the derivation of the standard form of PFT following Ref. [26]:One considers a system of N overdamped Brownian particles, where the i-th particleobeys the Langevin equation

γ~vLi (~rk, t) = ~Fi(~rk, t) + ~χi(t) (172)

64

with the friction constant γ, the single-particle velocities ~vLi that depend on the

particle positions ~ri and time t, the forces ~Fi, and the noises ~χi described by

Eqs. (73) and (74). Introducing the N -body density Ψ(~rk, t), the dynamics canequivalently be rewritten in the form of the Smoluchowski equation

∂

∂tΨ(~rk, t) = −

N∑i=1

~∇~ri · ~Ji(~rk, t) (173)

with the current of particle i

~Ji(~rk, t) = Ψ(~rk, t)γ−1(~Fi(~rk, t)− kBT ~∇~ri ln(Ψ(~rk, t)))= Ψ(~rk, t)γ−1 ~F tot

i (~rk, t)= Ψ(~rk, t)~vi(~rk, t),

(174)

where we have identified the total force ~F toti = ~Fi − kBT ~∇~ri ln(Ψ) on and the de-

terministic velocity ~vi = γ−1 ~F toti of the i-th particle. The deterministic velocity ~vi is

equivalent to the stochastic velocity ~vLi appearing in the Langevin equation (172) asfar as average values are concerned. This allows to introduce the one-body current(which can also be calculated using the stochastic velocity ~vLi ) as

~J(~r, t) =

⟨ N∑i=1

~vi(~rk, t)δ(~r − ~ri)⟩. (175)

We now have the necessary ingredients to set up PFT. The starting point is thegenerating functional

Rt[Ψ, ~vi] =

∫d3r1 · · ·

∫d3rN Ψ(~rk, t)

( N∑i=1

γ(1

2~vi(~rk, t)− ~vi(~rk, t)

)· ~vi(~rk, t) + U1(~ri, t)

) (176)

with the time derivative of the external potential U1 = ∂U1/∂t and the trial velocity

functions ~vi. (We follow the convention of Ref. [26] and use a subscript t to denotetime-dependence.) The actual velocity ~vi can then be determined from the variation

δRt[Ψ, ~vi]δ~vi(~rk, t)

= ~0 (177)

performed at fixed Ψ and t. Based on the generating functional (176), the free powerfunctional is defined as

Rt[ρ, ~J ] = min~vi→ρ, ~J

Rt[Ψ, ~vi], (178)

which denotes a minimization with respect to the trial velocities ~vi with the con-straint that a certain one-body density and current are realized (Levy method [613]).

65

After removing the dependence on the distribution function Ψ by using the many-bodycontinuity equation (173), we arrive at the simple variational principle

δRt[ρ, ~J ]

δ ~J(~r, t)= ~0. (179)

The power functional depends on density and current and also on the system’s history,i.e., it contains memory effects. In the variation, ρ(~r, t′) for t′ ≤ t and ~J(~r, t′) for t < t′

are held constant.By separating external contributions from the complete functional, one can obtain

the intrinsic free power

Wt[ρ, ~J ] = Rt[ρ, ~J ] +

∫d3r ( ~X(~r, t)− ~∇U1(~r, t)) · ~J(~r, t)− U1(~r, t)ρ(~r, t), (180)

where ~X(~r, t) contains all nonconservative forces. The intrinsic free power is furthersplit up as

Wt[ρ, ~J ] = Pt[ρ, ~J ] +

∫d3r ~J(~r, t) · ~∇

(δF

δρ(~r, t)− U1(~r, t)

)(181)

with the power dissipation functional Pt describing irreversible loss of energy and aterm for reversible contributions. The power dissipation can, moreover, be split intotwo contributions as

Pt[ρ, ~J ] = P idt [ρ, ~J ] + P exc

t [ρ, ~J ] (182)

with the ideal gas contribution

P idt [ρ, ~J ] =

∫d3r

γ ~J(~r, t)2

2ρ(~r, t)(183)

and the excess dissipation P exct [ρ, ~J ] resulting from particle interactions. If evaluated

at the DDFT current ~J = −Γρ~∇δF/δρ, Eq. (183) leads to the DDFT dissipationfunctional (226) (see Section 6.2). The variational principle (179) gives the fundamentalequation of motion

γ ~J(~r, t)

ρ(~r, t)+δP exc

t [ρ, ~J ]

δ ~J(~r, t)= −~∇ δF [ρ]

δρ(~r, t)+ ~X(~r, t). (184)

If one sets the excess dissipation to zero, one recovers DDFT as a special case. Theoriesthat include nonadiabatic contributions can be obtained by including effects of particleinteractions on the power dissipation. In practice, the application of PFT requires anapproximation for P exc

t [26]. Moreover, an accurate DDFT for the considered system

is needed [273], since the adiabatic DDFT term ~∇δF/δρ in Eq. (184) also has to beapproximated.

In Ref. [330], a reformulation of PFT based on velocity gradients ~∇⊗~v is developed.

A change of variables allows to express the power functional in terms of ~∇⊗ ~v. Thisis motivated by the need to construct approximations for the excess power functional

66

P exct . For example, the assumption that it can, to lowest order, be written as a nonlocal

bilinear in velocity gradients gives the form

P exct [ρ, ~J ] = kBT

∫d3r

∫d3r′

∫ t

0dt′ ρ(~r, t)(~∇⊗ ~v(~r, t))

:M4(~r − ~r′, t− t′) : (~∇⊗ ~v(~r′, t′))ρ(~r′, t′)

(185)

with the fourth-rank tensor M4 the velocity gradients are contracted with. With anadditional Markovian approximation and the assumption of spatial locality, this allowsto find, for the superadiabatic forces, the form familiar from the Stokes equation inhydrodynamics.

PFT has been generalized towards mixtures [27], Newtonian mechanics [29] (seeSection 5.2.3), quantum mechanics [28,526] (see Section 5.2.4), and active particles[296] (see Section 5.2.2). Methods for calculating the power dissipation are presentedin Ref. [527]. An iterative method for relating the one-body density and current to theexternal force generating them is described in Ref. [30]. Applications of PFT includethe calculation of correlation functions [31], the van Hove function [27], shear [329,331],demixing [332], and phase diagrams [32,300]. A further discussion of superadiabaticforces can be found in Refs. [287,340,528,529].

5.2.2. Nonspherical and active particles

A PFT for ABPs, which also takes orientational degrees of freedom into account,was derived by Krinninger and Schmidt [296]. The generating functional, which fortranslational motion is given by Eq. (176), has the form

Rt[Ψ, ~vi, ~vωi ] =

∫d3r1

∫S2

dΩ1 · · ·∫

d3rN

∫S2

dΩN Ψ(~rk, ~uk, t)(N∑i=1

γ(1

2~vi(~rk, ~uk, t)− ~vi(~rk, ~uk, t)

)· ~vi(~rk, ~uk, t)

+

N∑i=1

γω(1

2~vωi (~rk, ~uk, t)− ~vωi (~rk, ~uk, t)

)· ~vωi (~rk, ~uk, t)

).

(186)

Compared to Eq. (176), this generating functional includes the rotational velocities

~vωi , related to the total torque, and the corresponding trial fields ~vωi and frictionγω. The continuity equation for the one-body density

∂

∂t%(~r, ~u, t) = −~∇~r · ~J(~r, ~u, t)− ~∇~u · ~Jω(~r, ~u, t) (187)

contains two currents ~J and ~Jω for translational and rotational motion, respectively.Then, the variational equation (179), which is a generalized force balance, is supple-mented by the generalized torque balance

δRt[ρ, ~J, ~Jω]

δ ~Jω(~r, ~u, t)= ~0. (188)

67

The phase diagram of active particles is moreover calculated using PFT in Refs. [32,300].

5.2.3. Newtonian mechanics

Inertia can also be incorporated in PFT [29]. Here, the variational principle of PFT (see

Section 5.2.1) is formulated for ~J rather than for ~J . Denoting positions and momentaof the individual particles by ~ri and ~pi, respectively, and introducing the trialacceleration fields ~ai defined on phase space, one defines the functional

Gt =

∫d3r1

∫d3p1 · · ·

∫d3rN

∫d3pN

N∑i=1

(~Fi −m~ai)2

2mΨ(~ri, ~pi, t)−

∫d3r

m

2 〈ρ〉 〈˙~J〉

(189)

with the force ~Fi exerted on particle i, the phase-space distribution function Ψ, themany-body density ρ given by Eq. (47), and the total time derivative of the microscopic

current˙~J . A functional of the three fields ρ(~r, t), ~J(~r, t), and ~J(~r, t) can be obtained

by the constrained search

Gt[ρ, ~J, ~J ] = min~ai→ρ, ~J, ~J

Gt. (190)

Assuming physical values of ρ and ~J , the functional is minimized by the true value of~J , i.e.,

δGt[ρ, ~J, ~J ]

δ ~J(~r, t)= ~0. (191)

As in usual PFT, the functional Gt can be split into an ideal, external, and excesspart. Contributions that go beyond the Brownian case are contained in a “superpowerfunctional”.

5.2.4. Quantum mechanics

PFT has also been extended to quantum systems [28]. Since the Schrodinger equationis not overdamped, one needs to take inertia into account. Thus, the formalism isanalogous to the Newtonian PFT with inertia presented in Section 5.2.3 (which, in-terestingly, has been developed after quantum PFT). The variational principle is then

based on a functional Gt that also depends on the time derivative ~J of the current. Inthe quantum case, the force ~F and the current ~J are replaced by the correspondingoperators. Moreover, quantum systems are described using the many-particle wavefunction Ψ rather than the phase-space distribution Ψ. Hooke’s helium model atomwas explored using this method in Ref. [526].

68

Microscopic Hamiltonian dynamics

StochasticDDFT

Standard

DDFT

Additional

order

parameters

DDFT withmomentumdensity

DeterministicDDFT

stochastic deterministic

ExtendedDDFT

Functionalthermo-dynamics

interpolation withgeneral projections

micro-canonicalmethod,projectiononto ρ(~r, t)

canonicalmethod,projectiononto ρ(~r, t)

canonicalmethod,projectiononto ρi(~r, t)and e(~r, t)(or generalvariables)

canonicalmethod,projectiononto ρ(~r, t)and e(~r, t)

canonicalmethod,projectiononto ρ(~r, t)and ~g(~r, t)

Figure 5. Derivation of DDFT using the projection operator formalism.

5.3. Projection operator formalism

5.3.1. Overview

Projection operator methods are another central tool in the context of DDFT. Theseallow, in general, to derive coarse-grained equations of motion [530,531] for a set ofrelevant variables by projecting the complete microscopic dynamics onto the subdy-namics of the relevant variables. Based on the original work by Nakajima [33], Zwanzig[35], and Mori [34], a large class of such methods has been developed. These methodsall share the same general structure, but differ in certain details, such as the definitionof the projection operator or the domain of applicability.

The role of projection operators in the derivation of DDFT is visualized in Fig. 5.They can be classified based on whether they fix the value (“microcanonical” case)or the average (“canonical” case) of the relevant variables [276] (left of Fig. 5, seeSection 3.5.2 for a discussion of this point). Microcanonical methods allow to derivestochastic DDFT. The derivation of a deterministic DDFT with a free energy fromDFT requires a canonical method [15]. In these theories, the projection operator isgenerally time-dependent. They were developed by Robertson [377] and Kawasakiand Gunton [284], and subsequently generalized towards the dynamics of fluctuations[281] and time-dependent Hamiltonians [36,282]. A general overview is given in thetextbook by Grabert [279] and a simple introduction in Ref. [37]. Various names areused in the literature on DDFT, such as “Kawasaki-Gunton operator method” [15,277],“Zwanzig projection operator” [401], “Mori-Zwanzig-Forster technique” [40,212], and

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“Mori-Zwanzig formalism” [36]. (Some of these names, such as “Kawasaki-Guntonmethod”, refer to specific forms.)

As discussed in Section 3.3.4, projection operators can be used to derive DDFTby projecting onto the one-body density as a relevant variable [15,16,102,114,271,283]. Additional projections allow to go from DDFT to phase field crystal models[532]. When using the Mori theory, which is a simple form of the projection operatorformalism, one obtains a linear model that has different properties than full DDFT[469]. Yoshimori [533] used projection operators to extend DDFT to a rigid interactionsite model.

By choosing additional relevant variables, extensions of DDFT to further order pa-rameter fields can be (and have been) derived (right of Fig. 5 on the preceding page).This is done in extended dynamical density functional theory (EDDFT) for several con-centration fields for the different species of colloidal particles in mixtures, the energydensity [40], and the entropy density [212] (see Paragraph 5.3.2.1) and in functionalthermodynamics (FTD) for the energy density [138] (see Paragraph 5.3.2.2). For fluids,the momentum density can also be included [383,384,428,509,532,534,535]. Additionalorder parameters, such as an elastic strain field, can account for translational symme-try breaking in solids [511,532,536]. In Refs. [537,538], projection operators were usedto calculate elastic properties of solids (see Section 8.2.8). In principle, arbitrary vari-ables can be incorporated in DDFT, although for obtaining a Markovian equation itis required that these variables are slow compared to the microscopic dynamics. More-over, additional degrees of freedom can be incorporated by using a density Ψ(~r, ~p, t)that also depends on the momentum ~p [392] or a density %(~r, ~$, t) with the orientation~$ [401].

In addition, projection operator methods allow for important conceptual insightsinto DDFT. For example, they allow to connect deterministic and stochastic forms ofDDFT, thereby clarifying the origin and necessity of noise terms [277,535] (see Sec-tion 3.5.2). This is also useful for fluctuating hydrodynamics [382,383]. Furthermore,projection operators allow for an exact definition of the free energy functional [15].Finally, an interesting connection between projection-operator-based coarse grainingand numerical discretization has been found by de la Torre et al. [278].

5.3.2. Projection-operator-based extensions of standard DDFT

5.3.2.1. Extended dynamical density functional theory. As discussed in Sec-tion 5.3.1, the Mori-Zwanzig projection operator formalism allows to derive equationsof motion for arbitrary dynamical variables, which for slow variables can, in most cases,be approximated by memoryless dissipative transport equations. Since standard DDFTcan be derived by projecting the microscopic dynamics onto the number density as arelevant variable (see Section 3.3.4), an extended dynamical density functional theory(EDDFT) can be derived by projecting onto a larger number of variables.

The projection operator method allows, for a set of relevant variables with meanvalues ai(~r, t), to introduce a free energy F [ai] such that the dynamics is governedby the thermodynamic conjugates δF/δai(~r, t). This allows to derive the general trans-port equations (221) for conserved and (222) for nonconserved variables (see below),

along with microscopic expressions for the current ~Ji and the quasi-current Φi, respec-tively, that depend on the thermodynamic conjugates [40]. For close-to-equilibriumsystems, the current and quasi-current can be obtained from a dissipation functional[212] (see Section 6.2). Moreover, since the projection operator formalism is also astandard method for the derivation of MCT, this approach allows for insights into the

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relation of EDDFT and MCT [40].Wittkowski et al. [40] derived an EDDFT for particle mixtures by choosing the

concentration of each particle species ρi(~r, t) and the energy density e(~r, t) as rele-vant variables. The resulting theory is an extension of DDFT to colloidal mixturesand temperature gradients. In general, the expressions for the currents also containhydrodynamic interactions. Moreover, the entropy density (see Section 4.8.3) is in-corporated as a relevant variable into EDDFT in Ref. [212] within the framework oflinear irreversible thermodynamics, assuming that a local formulation of the first lawof thermodynamics holds.

For a general set of κ conserved relevant variables Ai(~r, t) that are fields onspacetime, the extended DDFT equations for the corresponding mean values ai(~r, t)read [40]

∂

∂tai(~r, t) = −~∇~r · Tr(ρ(t) ~Ji(~r, 0)) +

κ∑j=1

1

kBT~∇~r ·

∫d3r′D(ij)(~r, ~r′, t)~∇~r′

δF

δaj(~r′, t)

(192)

with the microscopic current ~Ji for the relevant variable Ai defined by the relation

iLAi(~r, t) = −~∇ · ~Ji(~r, t), (193)

where in classical mechanics the Liouvillian L is defined with the Poisson bracket ·, ·,and the diffusion tensor

D(ij)ml (~r, ~r′, t) =

∫ ∞0dt′ Tr

(ρ(t)(Q(t)Jj,l(~r

′, 0))eiLt′(Q(t)Ji,m(~r, 0))), (194)

where ρ(t) is the relevant probability density, Q = 1 − P is the complementary pro-

jection operator (see Section 3.3.4), and Ji,j = ( ~Ji)j . For real-valued relevant variables

with definite time-reversal signature, Onsager’s principle demands that D(ij)kl = D

(ji)lk .

A notable aspect of the general EDDFT equation (192) is the presence of the orga-

nized drift term ~∇~r ·Tr(ρ(t) ~Ji(~r, 0)). In usual DDFT, this drift vanishes by symmetry,which is a manifestation of the fact that DDFT describes overdamped dynamics. Forgeneral dynamical variables, this is, however, not guaranteed. In this case, the driftterm corresponds to the nondissipative part of the dynamics.

A notable feature of the EDDFT approach is that it, being based on the projectionoperator formalism, also provides dynamic equations for the correlation functions. Forthis reason, it is a natural framework for analyzing the relation of DDFT to MCT (seeRef. [40] and Section 6.1).

5.3.2.2. Functional thermodynamics. Functional thermodynamics (FTD), de-rived by Anero et al. [138], is another extension of DDFT towards nonisothermalsystems. As in EDDFT, the complete microscopic dynamics of a system is, using theprojection operator formalism, projected onto the number density and the energy den-sity as relevant variables. A conceptual aim of this framework is to study the relationof different “levels of description” of a certain system, which are characterized by thechoice of the relevant variables. Each level possesses its own entropy functional, whichcan be connected to that of other levels using a bridge theorem. This generalizes the

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theorem by Mermin [3] on the relation between density, external potentials, and freeenergy functional exploited in DFT (see Section 2.2).

5.4. Runge-Gross theorem

5.4.1. Quantum time-dependent density functional theory

Besides its importance in classical statistical physics, DFT is also an important tool ofmany-body quantum mechanics. Quantum DFT also has a dynamical extension, whichin contrast to its classical counterpart is usually referred to as “time-dependent densityfunctional theory” (TDDFT). (In this review, we use “DDFT” to refer to the classicaland “TDDFT” to refer to the quantum-mechanical method. Note that “TDDFT” issometimes also used in the literature to refer to classical DDFT.) Reviews of TDDFTcan be found in Refs. [614–622]. Quantum DFT is reviewed in Refs. [80,623–626]. Adiscussion of the relation of classical and quantum DFT can be found in Ref. [627].TDDFT is based on the Runge-Gross theorem, which is a formally exact result that,in a very similar form, can also be derived for classical systems [130]. From the latterresult, one can obtain classical DDFT as a limiting case. Here, we will first give abrief introduction to quantum DFT and TDDFT. The relation to classical DDFT isdiscussed in Section 5.4.2.

We follow Refs. [80,628]. The state of a system of N electrons can be describedusing the N -particle wave function Ψ(~r1, . . . , ~rN ), which satisfies the time-independentSchrodinger equation

HΨ = EΨ (195)

with the Hamiltonian H and energy eigenvalue E. The Hamiltonian can be written as

H = T + Vee + Vext (196)

with the kinetic energy T , the electron-electron interaction potential Vee, and an ex-ternal potential Vext. In position space, these are given by

T =

N∑i=1

− ~2

2me

~∇2~ri , (197)

Vee =

N∑i,j=1i 6=j

Q2e

8πε0

1

‖~ri − ~rj‖, (198)

Vext =

N∑i=1

U1(~ri) (199)

for a system of N electrons with mass me and charge Qe. Here, ~ = h/(2π) is thereduced Planck constant and ε0 is the vacuum permittivity. The ground state of thesystem is the state Ψ with the lowest energy E. In practice, Eq. (195) is impossibleto solve for all but the simplest systems. A more efficient description of many-particlesystems is possible in DFT: Just as classical DFT has allowed us to use the density ρinstead of the full phase-space distribution Ψ, quantum DFT allows to work with the

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electron density [80]

n(~r) = N

∫d3r2 · · ·

∫d3rN |Ψ(~r1, . . . , ~rN )|2 (200)

instead of the full N -body wavefunction. Since n(~r) is a function of 3 rather than 3Nvariables, this leads to a significant computational simplification. The reason why thisis possible is that, as shown by Hohenberg and Kohn [1], the external potential gen-erating a certain density n(~r) is unique up to a constant. Moreover, since the externalpotential determines the Hamiltonian which then fixes the ground-state wavefunctionvia the Schrodinger equation (195), the wavefunction is a functional of n(~r). The samethen holds for the internal energy of the system. One can therefore introduce theuniversal functional [1]

Funiv[n] = 〈Ψ |T + Vee|Ψ〉 , (201)

which is valid for any external potential. Here, 〈·〉 denotes a quantum-mechanicalexpectation value. Hohenberg and Kohn [1] showed that the ground-state energy canthen be found by minimizing the functional

E = Funiv[n] +

∫d3r n(~r)U1(~r). (202)

The minimization is over all nondegenerate densities that are v-representable, i.e.,that are ground-state densities of an external potential [80]. Moreover, the originalderivation by Hohenberg and Kohn [1] is restricted to nondegenerate ground states.A generalization was derived by Levy [613]. For the practical application of DFT,approximation methods were developed by Kohn and Sham [2]: We consider a nonin-teracting reference system that has the same density as the interacting system. For thenoninteracting system, the density can be written in terms of single-particle orbitalsζi(~r) as

ngs(~r) =

N∑i=1

|ζi(~r)|2, (203)

where the single-particle orbitals satisfy the Kohn-Sham equation(− ~2

2me

~∇2 + u(~r)

)ζi(~r) = Eiζi(~r) (204)

with the single-particle energies Ei. The potential u(~r) is the potential which pro-duces the ground-state density ngs(~r) in the noninteracting system and thus has afunctional dependence on n. Once the potential u(~r) is known, the ground-state en-ergy Egs can be calculated immediately. To determine this potential, we write it as

u(~r) = U1(~r) + uH(~r) + uxc(~r) (205)

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with the Hartree potential

uH(~r) =

∫d3r′

Q2e

4πε0

n(~r′)‖~r − ~r′‖ (206)

and the exchange correlation potential uxc(~r), which is not known exactly and has tobe approximated.

TDDFT extends quantum DFT by using a generalization of the Hohenberg-Kohntheorem which is known as the Runge-Gross theorem [81]. It states that for two N -electron systems that start from the same initial state Ψ0 and are subject to two differ-ent time-dependent potentials, the time-dependent densities will be different (unlessthe potentials only differ by a time-dependent function). The proof requires that thepotential can be Taylor-expanded in time. As in DFT, one therefore has a one-to-onecorrespondence between the density and the external potential. The main differenceto the time-independent case is that this correspondence holds for a fixed initial stateΨ0. Consequently, the external potential has functional dependence on both n andΨ0, where the latter dependence vanishes if the system starts in the ground state. Asin time-independent DFT, one can introduce a noninteracting reference system in anexternal potential u(~r, t) that reproduces the density n(~r, t) of the interacting system.The electrons in the noninteracting system satisfy the time-dependent Kohn-Shamequations

i~∂

∂tζi(~r, t) =

(− ~2

2me

~∇2 + u(~r, t)

)ζi(~r, t). (207)

Again, the potential u(~r, t) is decomposed in the form (205), this time with an ex-change correlation potential that depends on the density and on the initial state ofthe interacting system and the Kohn-Sham system.

5.4.2. Existence proof for DDFT

A TDDFT has been derived for classical systems by Chan and Finken [130], who usedan analogue of the Runge-Gross theorem to establish an invertible mapping betweenthe density ρ(~r, t) and the phase-space distribution function. For Hamiltonian systems,this allows to obtain a stationary action principle. The hydrodynamic behavior can,in analogy to quantum mechanics, be obtained from a Kohn-Sham-like noninteractingreference system. This framework allows to relate classical DDFT to quantum TDDFT,since it is derived along the same lines as the quantum mechanical formalism, but atthe same time contains classical DDFT as a limiting case. In a sense, one has therebyderived classical DDFT from the corresponding quantum-mechanical formalism (byformal analogy).

The proof by Chan and Finken [130] is often thought of as an existence proof for aclassical DDFT, but, unfortunately, it is nonconstructive [19,201,288,317]. It concerns asystem whose microscopic dynamics is described by Hamilton’s equations. For a fixedinitial phase-space distribution, solving the deterministic Liouville equation gives aunique mapping between the phase-space distribution and the external potential. Itis then shown that different potentials (that do not only differ by a purely time-dependent function) lead to different one-body currents, which allows, through thecontinuity equation, to obtain a unique mapping between the time-dependent one-bodydensity and the time-dependent external potential for a given initial distribution. This

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mapping also exists if the microscopic dynamics is Brownian rather than Hamiltonian.Most of the derivation in Ref. [130] is, however, restricted to Hamiltonian systems. Onecan introduce a noninteracting reference system that at each time reproduces the exactone-body density. The excess potential can be obtained as the functional derivative ofan “excess action functional” that in the simplest (adiabatic) case can be approximatedusing the equilibrium excess free energy. This allows to obtain the standard DDFTequation (35) [130]. In this approximation, memory effects are dropped.

6. Theories related to DDFT

In this section, we discuss the relation of DDFT to various other theories used in thetheoretical description of soft matter systems. This is important for understanding theprecise location of DDFT in the landscape of soft matter theory. On the one hand, thedevelopment of DDFT has historically been influenced by work on related methods,such as mode coupling theory (MCT) [114] or the Cahn-Hilliard equation [8,91,92].On the other hand, DDFT has influenced the development of other theories such asphase field crystal (PFC) models [44,135,136,629,630]. In addition to the historicalimportance, such a discussion also provides conceptual insights: Studying the relationof DDFT to MCT allows to understand to which extent DDFT can and cannot describeglassy systems, and comparing DDFT to general models of the gradient dynamics typeis helpful for work on spinodal decomposition and relaxation dynamics.

We start with a discussion of MCT (Section 6.1), which has been important atearly stages of the historical development of DDFT and continues to be important forthe study of the glass transition. Then, we continue with theories of the gradient dy-namics type, beginning with a general discussion of nonequilibrium thermodynamics(Section 6.2), which has also (in particular in the form of the Cahn-Hilliard equation)influenced DDFT. We subsequently turn to two younger theories (Sections 6.3 and 6.4)that have been influenced by DDFT. Here, PFC models (Section 6.3), which also havea gradient dynamics form, are of particular interest. Afterwards, we discuss nonequi-librium self-consistent generalized Langevin equations (Section 6.4), which are basedon nonequilibrium thermodynamics. Finally, we present dynamic mean-field theory(Section 6.5), which is a discrete model. The latter two approaches are not frequentlydiscussed in the literature on DDFT, but are nevertheless of conceptual interest andtherefore presented here for completeness.

6.1. Mode coupling theory

Mode coupling theory (MCT), reviewed in Refs. [631,632], is the most widely usedtheory for the description of the glass transition [633–635] (although it has its originin the theory of critical dynamics [636–638], see Ref. [115] for a discussion). Historically,MCT was very important for the development of DDFT, in particular in the formsproposed by Kirkpatrick and Wolynes [95] and Kawasaki [114]. There is a significantamount of literature discussing the relation of DDFT and MCT, in which, remarkably,two different views are expressed: In treatments of stochastic DDFT, MCT is typicallypresented as a limiting case of DDFT. DDFT is considered a more general theorythat models aspects of the dynamics of supercooled liquids and glasses not capturedby MCT [114,234,235]. The literature on deterministic DDFT, on the other hand,considers DDFT and MCT to be complementary theories that arise as different limitingcases of a general exact treatment [24,40]. After giving a brief introduction to MCT,

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we here discuss how this apparent contradiction can be resolved. Finally, we explainhow MCT can be derived from stochastic DDFT.

The object of interest in MCT is the normalized density correlator φ(~k, t), which isdefined as [235]

φ(~k, t) =〈ρ~k(t)ρ−~k(0)〉〈ρ~k(0)ρ−~k(0)〉 (208)

with the ensemble average 〈·〉 and the Fourier-transformed reduced density

ρ~k(t) =

∫d3r (ρ(~r, t)− ρ0)e−i~k·~r. (209)

The correlation approximately follows the closed equation of motion [131,235]

φ(~k, t) + νφ(~k, t) + Ω2(~k)φ(~k, t) = −∫ t

0dt′M(~k, t′)φ(~k, t− t′) (210)

with the friction coefficient ν, squared frequency Ω2(~k) = kBTk2/(mS(~k)), and mem-

ory kernel

M(~k, t) =kBTρ0

2(2π)3mk2

∫d3k′

(~k·~k′c(~k′)+~k·(~k−~k′)c(~k−~k′)

)2S(~k′)S(~k−~k′)φ(~k′, t)φ(~k−~k′, t),

(211)

where S(~k) is the static structure factor and c(~k) is the Fourier-transformed directpair-correlation function. Note that the explicit form of the MCT equation can bedifferent and depends on the system (e.g., on whether we consider simple or colloidalfluids) [40].

Using certain approximations, this result can be derived using the projection oper-ator formalism (see Section 5.3.1). The derivation can be found in Ref. [632]. Startingfrom the microscopic dynamics, a formally exact equation of motion for the correla-tion function is obtained. This equation contains a memory term, in which the memorykernel is the autocorrelation of the random force. The random force corresponds to thepart of the complete dynamics that is orthogonal to the relevant slow variables givenby the collective density modes. Next, one makes the approximation that the randomforce is dominated by products of two density modes, which corresponds to the sim-plest nontrivial form. (Note that, due to the nature of the linear projection operator,products of the density modes are contained in the random force if we project onto thedensity modes.) Finally, it is assumed that the four-point density correlation functionin the memory kernel, which one gets as a result, can be approximated by a prod-uct of two-point correlation functions. This gives the “MCT approximation” for thememory kernel [639]. (Note that the name “MCT approximation” is used with differ-ent meanings in the literature. Here, we use it to denote the approximations requiredfor obtaining the MCT expression (211) from the full density dynamics described bystochastic DDFT, as discussed below and visualized in Fig. 3 on page 34.)

The success of MCT thus shows that, if the density is a slow variable, the samecan be assumed about its products (which appear in the random force and thereforein the memory kernel). This suggests that, in a system with the density as a slowvariable, a more general theory can be derived by projecting the microscopic dynamics

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of the system onto the set of all nonlinear functions of the density [235], which isprecisely what is done in the derivation of (stochastic) DDFT in the (microcanonical)projection operator formalism [102]. The assumption of a slow density is motivated bythe fact that, in dense liquids, particle motion is restricted, whereas momentum canbe transferred very quickly [234].

Physically, MCT has the interesting feature that it predicts a transition from anergodic to a nonergodic state. This is related to its role as a theory of the glasstransition: Glasses form when a liquid is cooled below its freezing temperature, butdoes not freeze into a crystal, which would be the thermodynamic equilibrium state(“equilibration”), but into some other state [235]. Systems in glassy states have alocal structure that looks like that of a fluid. However, the viscosity is much larger.The formation of glasses has a kinetic rather than a thermodynamic origin [140],which is reflected by the fact that their dynamical behavior depends strongly on theirhistory and the way they were prepared (“aging”). A simple picture for the origin ofglass formation is that, in dense supercooled liquids, particles get trapped in cagesand thus cannot move to their equilibrium positions. These geometrical constraints,which prevent the system from accessing the complete phase space, make the systemnonergodic.

However, the actual physics of the glass transition is more complicated. Real mate-rials are always subject to thermal fluctuations and these fluctuations have the con-sequence that there is no ideal ergodic-to-nonergodic transition. To overcome thisproblem, extensions of MCT have been developed (see Ref. [631] for an overview).The shortcomings of the simple MCT motivated the development of the stochasticDDFT by Kawasaki [114] (see Section 3.2.2), in which thermal fluctuations are explic-itly included. Stochastic DDFT allows for the derivation of MCT as a limiting case.Details are discussed below.

A somewhat different view on the relation of DDFT and MCT can be found in theliterature on deterministic DDFT, where DDFT and MCT are viewed as different limit-ing cases of a general exact description. For example, Wittkowski et al. [40] compared,within the framework of EDDFT (see Paragraph 5.3.2.1), the derivation of DDFTand MCT via the projection operator formalism (see Section 3.3.4) and noted thatthe derivations involve different approximations. In (E)DDFT, it is assumed that theensemble-averaged one-body density (and possible additional variables) follow closedmemoryless transport equations (Markovian approximation). In MCT, on the otherhand, one makes a linearization, but does not assume the dynamics to be Markovian(since MCT contains a memory term). A similar conclusion was reached by Brader andSchmidt [24] within the framework of PFT (see Section 5.2), who argued that DDFT isobtained by neglecting superadiabatic contributions to the power dissipation, whereasMCT corresponds to the assumption that these contributions have a certain form.

While this is certainly a topic that deserves further investigation, it should be notedhere that the views “stochastic DDFT allows to derive MCT” and “deterministicDDFT and MCT are complementary” do not necessarily have to be in conflict. Tosee this, consider the DDFT equation (50) obtained by Dean [105]. This is an exactdescription of a system of Brownian particles, from which the deterministic DDFTcan, as shown by Marini Bettolo Marconi and Tarazona [12], be derived with addi-tional approximations. On the other hand, since MCT for colloidal fluids can also bederived from the system of Langevin equations that is equivalent to Eq. (50), it followsthat MCT and deterministic DDFT are complementary approaches that both arise aslimiting cases of the (exact) stochastic DDFT by Dean.

An additional point that needs to be taken into account is that, as discussed in

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Sections 3.2.2 and 3.5.2, stochastic DDFT can be formulated for the exact microscopicdensity operator (47) (defined as a sum over Dirac delta distributions) or for a coarse-grained density. This also affects an MCT derived from these theories. If the stochasticDDFT describes a coarse-grained density, this will also be the density that entersthe correlator (208) (an example is the derivation by Archer [131] presented below).Kawasaki [114] explicitly compares his derivation of MCT from stochastic DDFT tothe derivation of MCT from the Smoluchowski equation by Szamel and Lowen [640]and notes as a crucial difference that the former employs a continuum description. Aderivation of an MCT for the correlator (208) with ρ given by the microscopic densityoperator ρ is possible by starting from a DDFT for the microscopic density [213]. Therelations between the different forms of DDFT and MCT are visualized in Fig. 3 onpage 34.

The problems regarding the relation to MCT discussed in the literature on deter-ministic DDFT do not necessarily apply to derivations from stochastic DDFT. Asdiscussed by Wittkowski et al. [40], Archer’s result concerns the temporally coarse-grained density, while the derivation of MCT from the projection operator formalismstarts from a microscopic theory. In the literature on deterministic DDFT, “MCT”typically denotes a dynamic equation for the correlator of the microscopic density while“DDFT” denotes an adiabatic transport equation for the ensemble-averaged density.This then leads to an incompatibility that is not necessarily present if “MCT” and“DDFT” are understood in a different way.

One of the problems in combining DDFT and MCT, arising in the microscopicderivation via the projection operator formalism, is that DDFT neglects memory ef-fects while MCT does not [40]. As discussed in detail in the textbook on projectionoperators by Grabert [279], the question whether or not memory effects in the trans-port equations for the macroscopic variables can be neglected depends on how largethe set of macroscopic variables is. In particular, a Fokker-Planck equation (as arisingin stochastic DDFT) for a dynamical variable can be Markovian even if the equationfor its mean value (as given by deterministic DDFT) contains memory effects. In thederivation of such a Fokker-Planck equation, one projects onto all nonlinear functionsof the relevant variables and assumes these to be slow [279]. In contrast, the deriva-tion of MCT uses only the density itself as a relevant variable, such that quadraticfunctions of the density contribute to the memory kernel. If these are also slow, it is,of course, inappropriate to ignore the memory effects, while it still can be appropriatefor a Fokker-Planck equation that includes the quadratic functions.

Regarding the PFT-based argument, it is important to note that not all forms ofstochastic DDFT employ the adiabatic approximation of deterministic DDFT. Since,by the terminology of PFT, everything that is not included in the adiabatic dynamicsof deterministic DDFT is a “superadiabatic force”, it is thus possible that stochasticDDFT allows to model phenomena that would be classified as “superadiabatic” inPFT. Consequently, the positions of Kawasaki [114], Archer [131], Wittkowski et al.[40], and Brader and Schmidt [24] are compatible, even though they superficially ap-pear to be incompatible.

In addition, DDFT and MCT can be combined in the microscopic calculation ofcertain properties of complex fluids. This has been done, e.g., for ionic systems [117–120,406,641–643], friction in supercooled liquids [644,645], and orientational relaxation[552,553]. We discuss the case of ions in Section 8.1.10.

The aforementioned derivation of MCT from stochastic DDFT can be done in atleast three ways. First, it was, with certain approximations, derived by Kawasaki[114] using projection operators (see also Ref. [646]) via a similar procedure as in the

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derivation by Szamel and Lowen [640] from the Smoluchowski equation. Second, it can,with certain approximations, be obtained by renormalized perturbation theory at one-loop order [261]. Mode coupling effects here take the form of a lifetime renormalization[233,631]. The full one-loop-order result, however, goes beyond MCT, and additionalapproximations are required to recover it [213] (see Section 7.2.3). A third derivation,performed by Archer [131] using ideas by Kawasaki [219] and Zaccarelli et al. [647],employs DDFT for atomic fluids (see Section 4.7.1) and is presented here. It is lessrigorous than the other types of derivation, but also much simpler. Starting from aDDFT for the coarse-grained density, it gives an MCT equation for the correlator ofthe coarse-grained density.

One starts from the exact microscopic dynamics and then performs a temporalrather than a spatial coarse graining. The temporally coarse-grained density is

ρ(~r, t) =

∫ ∞−∞dt′Kt(t− t′)ρ(~r, t′), (212)

where Kt(t) is a normalized function that has finite width. After temporal coarsegraining, one obtains an equation of the form (146). Making the approximation B ≈ νρleads to an equation that looks like Eq. (148). (Strictly speaking, a dynamic equationfor the coarse-grained density should contain thermal noise [211]. However, thermalnoise is not relevant for MCT [219].) Performing a Taylor expansion of the free energyin ∆ρ(~r, t) = ρ(~r, t)− ρ0 and then a spatial Fourier transformation gives

ρ~k(t) + νρ~k(t) + Ω2(~k)ρ~k(t) = R(~k, t), (213)

where ρ~k(t) is the Fourier transformation of ∆ρ(~r, t), Ω2(~k) = kBTk2/(mS(~k)) with

the static structure factor S(~k) = 1/(1− ρ0c(~k)), c(~k) is the Fourier transformation ofthe direct correlation function, and

R(~k, t) =kBT

(2π)3m

∫d3k′ ~k · ~k′ρ~k′(t)c(~k

′)ρ~k−~k′(t). (214)

This is compared to the general form (210) that can be obtained from the projection

operator formalism. The memory kernel M(~k, t) depends on the time correlation ofthe random force. Assuming that one can use R, given by Eq. (214), (rather than theexact random force obtained from the projection operator formalism) for calculating

the memory kernel M(~k, t), and making the usual factorization approximation (seeSection 6.1), gives

M(~k, t) =kBTρ0

2(2π)3mk2

∫d3k′ (~k·~k′c(~k′)+~k·(~k−~k′)c(~k−~k′))2S(~k′)S(~k−~k′)φ(~k′, t)φ(~k−~k′, t),

(215)which is the MCT expression (211) (although derived here for the correlator of thecoarse-grained density). Another discussion of the relation between DDFT and MCTcan be found in Ref. [504].

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6.2. Nonequilibrium thermodynamics

The general structure of DDFT is shared by a larger class of close-to-equilibriumdynamical theories known as “gradient dynamics”. Here, the time evolution of (a setof) conserved or nonconserved variables is driven by a thermodynamic potential. Ageneral form of gradient dynamics for a conserved variable ac is

∂

∂tac(~r, t) = ~∇ ·

(Mg(ac(~r, t))~∇ δF [ac]

δac(~r, t)

)(216)

with the positive mobility function Mg [648–652]. If ac is a vector containing multiplevariables, Mg is a matrix that is positive definite [546,648,653]. The DDFT equation(35) is a special case of Eq. (216) corresponding to ac = ρ and Mg = Γρ. Theories ofthis form can, in nonequilibrium thermodynamics, be derived from Onsager’s principle[654,655]. Onsager’s linear thermodynamics can also be used to obtain the DDFTequation [656]. A further discussion of Onsager’s variational principle can be foundin Refs. [657–659]. The structure of gradient dynamics thus relates DDFT to othernonequilibrium theories [546,660–663].

A very simple example is the Cahn-Hilliard equation describing the dynamics of aconserved order parameter field ϕch(~r, t), which is obtained from Eq. (216) by using aconstant mobility function Mch and a free energy [92]

F =

∫d3r

(κch

2(~∇ϕch)2 + fch(ϕch)

), (217)

where κch > 0 is a constant. The free energy (217) consists of a term penalizinggradients and the free energy density fch of a homogeneous solution, which is oftenapproximated as a fourth-order polynomial [651,664]. Combining Eqs. (216) and (217)gives the dynamic equation [91]

∂

∂tϕch = Mch

~∇2

(∂fch(ϕch)

∂ϕch− κch

~∇2ϕch

). (218)

Equation (218) is a simple model equation for spinodal decomposition [91]. Asis easily confirmed by a linear stability analysis (see Paragraph 7.2.1.1), a homo-geneous state ϕch(~r, t) = ϕch,0, where ϕch,0 is a constant, becomes unstable for(∂2fch/∂ϕ

2ch)|ϕch=ϕch,0

< 0. This result has inspired later work on spinodal decom-position in DDFT [8,14], which is discussed in Paragraph 7.2.1.1 and Section 8.2.2.Similar methods have also gained importance for other types of soft and active mattermodels [557,665–667].

Within linear irreversible thermodynamics, equations of motion can be derived froma dissipation functional. This was discussed by Wittkowski et al. [212] for the case of(extended) DDFT. For general conserved variables ac

i (~r, t) and nonconserved variablesani (~r, t), the thermodynamic forces are given by

~ac]i (~r, t) = −~∇δF [ak]

δaci (~r, t)

, (219)

an]i (~r, t) =

δF [ak]δani (~r, t)

. (220)

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Here, the superscript ], not to be confused with the related symbol \ introduced inSection 3.3.4, denotes a thermodynamic force, which has a different form for conservedand nonconserved variables. The equations of motion for the relevant variables aregiven by

∂

∂taci (~r, t) + ~∇ · ( ~Ji,R(~r, t) + ~Ji,D(~r, t)) = 0, (221)

∂

∂tani (~r, t) + Φi,R(~r, t) + Φi,D(~r, t) = 0 (222)

with the currents ~Ji and quasi-currents Φi that have a reversible and a dissipativecontribution, denoted by subscripts R and D, respectively. From the dissipation func-tional

R[ai, a]i] =

∫d3r r(~r, t), (223)

giving the energy dissipated per unit time and defined by the integral over the dis-sipation function r(~r, t), one can obtain the dissipative currents and quasi-currentsas

~Ji,D(~r, t) =δR[ai, a]i]δ~ac]i (~r, t)

, (224)

Φi,D(~r, t) =δR[ai, a]i]δan]i (~r, t)

. (225)

General expressions for the dissipation function are derived in Ref. [212]. In particular,the standard DDFT equation (35) can be obtained from the dissipation functional

R[ρ, ~ρ]] =

∫d3r

1

2Γρ(~r, t)(~ρ](~r, t))2, (226)

which, together with Eqs. (219), (221), and (224), constitutes a reformulation of

DDFT. The thermodynamic force, defined by Eq. (219), is given by ~ρ] = −~∇δF/δρ.A further discussion of the relation of DDFT-based models to linear irreversible ther-modynamics can be found in Ref. [556].

6.3. Phase field crystal models

Phase field crystal (PFC) models are another example of theories with a gradientdynamics structure. They arise as a limiting case of DDFT upon assuming a constantmobility and making certain approximations for the free energy [370] (see below fora detailed discussion). Note that the line of demarcation between DDFT and PFC isdrawn in different ways by different authors, sometimes models with a nonconstantmobility are also called “PFC models” [492]. In standard PFC models, a system isdescribed in terms of a dimensionless conserved order parameter field ψ(~r, t) governedby the dynamic equation

∂

∂tψ = D~∇2 δF

δψ. (227)

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PFC models were first proposed phenomenologically [135,629,630] and then connectedto DFT [136] and DDFT [44]. One can give PFC models a microscopic foundation byderiving them as an approximation to DDFT. This allows to connect the parametersof PFC models to the parameters of the microscopic dynamics [668,669] and to givea physical interpretation to the PFC order parameter field ψ [670] by identifying itas a dimensionless form of the deviation of the density ρ from a reference value. Inthis sense, PFC models are an intermediate theory between atomistic models (such asDDFT) and macroscopic continuum theories [671,672]. PFC models are less accurate,but also simpler. (Some authors classify PFC models as a form of DDFT [364]. We donot adapt this terminology here.)

Applications of PFC models, such as pattern formation or solidification, frequentlyoverlap with those of DDFT. This allows for a comparison of both methods. An ex-ample is the study of vacancy diffusion in colloidal crystals by van Teeffelen et al.[347], who compared results of both DDFT and PFC models to BD simulations. Itwas found that DDFT correctly predicts the temperature dependence of the diffusionconstant, whereas modifications are required in PFC models.

DDFTPerturbative

DDFTPFC

models

functionalTaylor

expansiongradientexpansion

Figure 6. Derivation of PFC models from DDFT.

The derivation of PFC models is visualized in Fig. 6. Following Ref. [139], we intro-duce the order parameter field of PFC models ψ as a dimensionless parametrizationof deviations from a constant reference density ρ0 in the form

ρ(~r, t) = ρ0(1 + ψ(~r, t)). (228)

This parametrization is inserted into the free energy (10), which allows for three mainapproximations [370]:

(1) Assuming small deviations from the reference density, a functional Taylor ex-pansion of F in ψ is performed.

(2) A Ramakrishnan-Yussouff or random phase approximation is made for the freeenergy functional.

(3) The functional is made local using a gradient expansion.

The Taylor expansion of the ideal gas free energy gives (dropping irrelevant con-stants)

Fid[ψ] = ρ0kBT

∫d3r

(ψ +

ψ2

2− ψ3

6+ψ4

12

), (229)

where the expansion is performed up to fourth order to allow for the formation of stablecrystals. The excess free energy has, after performing the Ramakrishnan-Yussouff ap-proximation, the form of an integral over a convolution. To make this functional local,we assume translational and rotational invariance to write the direct pair-correlation

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function as

c(2)(~r1, ~r2) = c(2)(‖~r1 − ~r2‖) = c(2)(r) (230)

and then, exploiting the symmetries, perform a Taylor expansion of the Fourier trans-formed direct pair-correlation function c(~k) around the wavevector ~k = ~0 in the form

c(~k) = c0 + c2k2 + c4k

4 (231)

with expansion coefficients c0, c2, and c4. Again, we have stopped the expansion atfourth order. After an inverse Fourier transformation, the expansion in wavevectors~k becomes an expansion in gradients. Omitting irrelevant constants, the excess freeenergy becomes

Fexc[ψ] = −ρ0

2kBT

∫d3r (A1ψ

2 +A2ψ~∇2ψ +A3ψ~∇4ψ), (232)

where the expansion coefficients Ai are moments of the direct pair-correlation func-tion. Combining Eqs. (229) and (232) and rescaling gives the complete PFC free energy

Fpfc[ψ] = ρ0kBT

∫d3r

(A′1ψ2 +A′2ψ~∇2ψ +A′3ψ~∇4ψ − ψ3

6+ψ4

12

)(233)

with the rescaled expansion coefficients A′i. Making the further approximation thatthe mobility is constant and equal to Dρ0 gives the equation of motion

∂

∂tψ = D~∇2 δFpfc[ψ]

δψ(234)

for the PFC model.The approximations made in the derivation of PFC models from DDFT have some

consequences for the structure and predictions of the models. These were discussed indetail by Archer et al. [370]. They identify dropping a term of the form

~∇ · (ψ~∇(Lψ)) (235)

with the nonlocal operator L defined by Lψ(~r, t) = −ψ(~r, t)+ρ0

∫d3r′ c(2)(~r, ~r′)ψ(~r′, t),

along with the assumption of constant mobility and an expansion of the logarithm inthe ideal gas free energy, as the central approximation that sets the border betweenDDFT and PFC models. The reason is that, after making the substitution ρ = ρ0(1 +ψ), performing a functional Taylor expansion of the free energy, truncating at fourthorder, and taking zero-wavelength components of third- and fourth-order correlationfunctions (following the derivation of PFC models by Huang et al. [456]), one obtains

∂

∂tψ = −~∇2(Lψ + Caψ

2 + Cbψ3 + Ccψ

4)− ~∇ · (ψ~∇(Lψ)), (236)

where Ca, Cb, and Cc are constants. Dropping the term (235) in Eq. (236) gives an

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equation of the form

∂

∂tψ = ~∇2 δF [ψ]

δψ(237)

with some functional F [ψ]. Deriving an equation of the form (237) from DDFT requiresthe assumption of a constant mobility. However, this assumption is incompatible withan ideal gas free energy of the form (11), since it would lead to

∂

∂tρ(~r, t) = D~∇ ·

(1

ρ(~r, t)~∇ρ(~r, t)

)(238)

instead of the usual diffusion equation (37) in the noninteracting case (the factor ρfrom the mobility would usually cancel the 1/ρ resulting from the logarithm). Hence,dropping (235) is necessarily connected to the constant mobility assumption and theexpansion of the logarithm. This expansion does not come without a price: It leads to asecond spinodal point in the phase diagram, such that PFC models have an unphysicalphase behavior. It predicts that structures formed at high densities melt again to auniform state at even higher densities.

In Ref. [199], DFT and PFC approaches are compared at various stages of theapproximation using hard-sphere simulations. Tupper and Grant [673] interpreted PFCmodels as the result of coarse graining molecular dynamics in time. A further discussionof the coarse-graining chain leading from microscopic dynamics to DDFT and then toPFC models using the projection operator method (see Section 5.3.1) can also be foundin Ref. [532]. The derivation of PFC models is moreover presented in Refs. [674–676].

In Refs. [380,456], a binary PFC model is derived from DDFT. Lowen [492] hasderived a PFC model for liquid crystals (consisting of particles with orientationaldegrees of freedom) from the corresponding DDFT. The derivation of an eighth-orderPFC model can be found in Ref. [677]. Praetorius and Voigt [421] derived an advectedPFC model from an advected DDFT. The dissipative dynamics of orientable particleswas moreover derived by Wittkowski et al. [556] based on DDFT and compared tomacroscopic Ginzburg-Landau theories, building upon the DFT-based discussions inRefs. [585,678]. The nematic tensor Qij can also be included as an order parameterin PFC models [679,680]. An “anisotropic phase field crystal model” for condensedmatter systems built from oriented nonspherical particles, proposed by Prieler et al.[681] (see also Ref. [682]), was derived from DDFT by Choudhary et al. [494].

Finally, PFC models can also be developed for active particles. The first theory ofthis type, proposed by Menzel and Lowen [683], was derived from active DDFT (seeSection 4.6.3) by Menzel et al. [493]. It reads

∂

∂tψ = ~∇2 δF

δψ− v0

~∇ · ~P , (239)

∂

∂t~P = ~∇2 δF

δ ~P−DR

δF

δ ~P− v0

~∇ψ (240)

with the self-propulsion velocity v0, polarization ~P (~r, t) arising from an orientationalexpansion (see Section 4.6.1), and rotational diffusion constant DR. Microscopically,the system of active particles is described by a DDFT for the orientation-dependentdensity %(~r, ~u, t) (see Sections 4.6.1 and 4.6.3). The derivation of Eqs. (239) and (240)from the microscopic DDFT for active particles employs a Cartesian expansion of the

84

density % in the form (131) (see Section 4.6.1), which gives ψ and ~P as zeroth and firstorders, respectively. In future work, this derivation could be extended to active PFCmodels on a sphere [684] starting from a DDFT for particles on a spherical surface[454]. A further discussion of active PFC models can be found in Refs. [685–688].

6.4. Nonequilibrium self-consistent generalized Langevin equations

The theory of nonequilibrium self-consistent generalized Langevin equations (NE-SCGLE), derived by Ramırez-Gonzalez and Medina-Noyola [689], is a nonequilibriumgeneralization of the self-consistent generalized Langevin equations (SCGLE) [690] forcolloids. It is derived from a nonequilibrium extension of the Onsager theory [691] anddescribes the relaxation dynamics of colloidal systems based on the diffusion equation

∂

∂tρ(~r, t) =

D

kBT~∇ ·(b(~r, t)ρ(~r, t)~∇µ(~r, [ρ])

)(241)

for the local concentration ρ with the local reduced mobility b(~r, t) describing localfrictional effects. The standard DDFT equation (35) is recovered when setting b(~r, t) =1. Equation (241) is coupled to the equation

∂

∂tσc(~r, ~r

′, t) = D~∇~r ·(ρ(~r, t)b(~r, t)~∇~r

∫d3r′′ ε(~r, ~r′′, [ρ])σc(~r

′′, ~r′, t))

+D~∇~r′ ·(ρ(~r′, t)b(~r′, t)~∇~r′

∫d3r′′ ε(~r′, ~r′′, [ρ])σc(~r

′′, ~r, t))

− 2D~∇~r ·(ρ(~r, t)b(~r, t)~∇~rδ(~r − ~r′)

)(242)

for the covariance σc(~r, ~r′, t) with the thermodynamic matrix

ε(~r, ~r′, [ρ]) =1

kBT

δµ(~r, [ρ])

δρ(~r′, t). (243)

NE-SCGLE theory has also been extended to multicomponent systems [692]. It ismainly used for glass-forming liquids, allowing to describe both equilibration [693]and aging [694] and in particular the crossover from the equilibration to the agingregime [695,696]. Moreover, it has been applied to arrested spinodal decomposition[697].

6.5. Dynamic mean field theory

A discrete theory that has a close relationship to DDFT is dynamic mean field theory(DMFT) or mean-field kinetic theory (not to be confused with the mean-field dy-namical density functional theory [17] or dynamic mean-field density functional theory[124]). This method, based on work by Martin [698] and Penrose [699], describes thedynamics of particles on a lattice. It has been interpreted as a form of DDFT [700,701],which is motivated by formal analogies discussed below. The “time-dependent densityfunctional theory” for lattice gases [106,292,702–707], which has been developed asa time-dependent extension of classical DFT, can also be seen as a form of DMFT[708]. Applications of DMFT include fluids in porous materials [700,709–715], wet-ting [701,716,717], evaporation [718], diffusion [719,720], and gelation [721]. Moreover,

85

DMFT has been extended to hydrodynamic interactions [402]. A review of latticemethods of this type can be found in Ref. [704]. Here, we focus on the relation toDDFT, which is discussed for DMFT in Ref. [700]:

Lattice DFT can be used to describe equilibrium states of fluids based on a freeenergy F (ρk) that depends on the mean density at the individual lattice sites in themean-field approximation. We denote by ρi the mean density at lattice site i [722] andassume a 1D lattice for simplicity. The exchange dynamics conserves the total particlenumber (Kawasaki dynamics [723]), such that the dynamics is given by

∂

∂tρi(t) = −

∑j=i±1

Jij(t) (244)

with Jij being the net flux from site i to site j. In the mean-field approximation, theflux is given by

Jij = wijρi(1− ρj)− wjiρj(1− ρi) (245)

with the transition probability wij . Defining the (symmetric) mobility as

Mij = − Jijµj − µi

(246)

with the chemical potential µi corresponding to ρi, the flux can be written as

Jij = −Mij(µj − µi). (247)

Inserting Eq. (247) into Eq. (244) gives

∂

∂tρi =

∑a=±1

Mi,i+a(µi+a − µi)

=1

2

∑a=±1

(Mi,i+a(µi+a − µi) +Mi,i−a(µi−a − µi)

),

(248)

where we have assumed the lattice to have a center of symmetry. With theforwards/backwards-difference operators Z±a, defined as

Z±afi = ±fi±a ∓ fi, (249)

Eq. (248) can be written as

∂

∂tρi =

1

2

∑a=±1

Z−aMi,i+aZaµi =1

2

∑a=±1

Z−aMi,i+aZa∂F

∂ρi, (250)

which has the form of a DDFT equation (the free energy F is defined in discretizedform).

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7. Analyzing a DDFT

After having presented the construction of DDFT (Section 3), its extensions (Sec-tions 4 and 5), and its relation to other theories (Section 6), we now proceed to themore practical problems of solving the dynamic equation(s) of DDFT. This will be(roughly) done in order of increasing generality: First, steady solutions are discussedin Section 7.1. Second, we introduce dynamics in Section 7.2 by explaining systematicperturbative expansions. Third, numerical methods are presented in Section 7.3, whichallow to study the full dynamics of DDFT.

7.1. Steady solutions

In many cases, the simplest way to study a dynamic equation is to consider its steady(in the simplest case stationary) solutions. In the case of the deterministic DDFT equa-tion (35), which describes the ensemble-averaged one-body density (see Section 3.5.2),the stationary solution – at least for an isolated system – is given by the solutions of theDFT equation (8). This reflects the fact that, as discussed in Section 3.5.4, DDFT de-scribes the approach to thermodynamic equilibrium. DFT, which is discussed in detailin Section 2.2, describes the equilibrium state of the ensemble-averaged density andis therefore the equilibrium solution of deterministic DDFT. For a stochastic DDFTsuch as Eq. (40), which does not describe the ensemble-averaged density, the station-ary solution is given by an equilibrium distribution of the form P [ρ] ∝ exp(−βF [ρ])[105].

However, this is not the most general case, since driven systems do not generallyapproach equilibrium. In driven systems, steady states can therefore arise that donot correspond to equilibrium states. A simple and illustrative example was discussedby Penna and Tarazona [724]: Consider (in 1D) a potential of the form V (x, t) =V (x − vt) = V (x) with a speed v and the position in a co-moving frame x = x − vt.We write the density ρ as ρ(x−vt) and look for a stationary solution in the co-movingframe. In this case, using ∂tρ = −v∂xρ, Eq. (35) can be integrated twice to give

δF [ρ]

δρ(x)+v

Γ(x− x0)− Jint

Γ

∫ x

x0

dx′1

ρ(x′)= µ, (251)

where x0 is the (arbitrary) integration boundary, µ an integration constant related tothis boundary, and Jint an integration constant with the dimension of a current. Forv = 0, Jint has to vanish and Eq. (251) reduces to the DFT equation (8), such thatthe only solution for the stationary density is the equilibrium density. In this case, µ isthe chemical potential of DFT. This, however, is not the case for v 6= 0. In Ref. [291],a similar method is applied to a phase-space description, which shows that, althoughthere is a general agreement with the DDFT results, inertia can lead to additionaleffects such as the formation of a wake.

More generally, a steady state of a DDFT with a time-dependent external force doesnot have to be a stationary state, i.e., the system might relax not to a time-independentstate described by static DFT but to, e.g., an oscillatory state [19,41].

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7.2. Perturbative approaches

7.2.1. Linearization

7.2.1.1. Linear stability analysis. A useful analytical method for the analysis ofphase separation in DDFT is the linear stability analysis. Its application to spinodaldecomposition in DDFT was first suggested by Evans [8]. Here, we present it followingRef. [14]: One writes the density as

ρ(~r, t) = ρ0 + ∆ρ(~r, t), (252)

where ρ0 is the density of the homogeneous reference state and ∆ρ(~r, t) is the deviation.If the homogeneous state is unstable, then a deviation will grow in time. This can bedetermined by inserting the parametrization (252) into Eq. (35). If we assume thatthe deviation is small, i.e., if we consider the initial stage of decomposition, then wecan linearize around the homogeneous state to find

1

ΓkBT

∂

∂t∆ρ(~r, t) = ~∇2∆ρ(~r, t)− ρ0

~∇2

∫d3r′ c(2)(‖~r − ~r′‖; ρ0)∆ρ(~r′, t), (253)

where the direct pair-correlation function c(2)(~r, ~r′) takes the form c(2)(‖~r− ~r′‖; ρ0) =c(2)(r, ρ0) for a homogeneous fluid of spherical particles. We introduce the Fouriertransformed density deviation as

ρ~k(t) =

∫d3r ei~k·~r∆ρ(~r, t) (254)

with the wavevector ~k, where the notation ρ~k(t) with ~k as a subscript (which we useonly for the density) is adapted from the MCT literature. A Fourier transformationof Eq. (253) gives

1

ΓkBT

∂

∂tρ~k(t) = (−k2 + ρ0k

2c(k))ρ~k(t) (255)

with the wavenumber k = ‖~k‖ and Fourier-transformed direct pair-correlation function

c(k) =∫

d3r exp(i~k · ~r)c(2)(r, ρ0). This is a linear differential equation for ρ~k(t) thathas the solution

ρ~k(t) = ρ~k(0) exp(λ(k)t) (256)

with the initial deviation ρ~k(0) and the dispersion relation λ(k) = −ΓkBTk2(1 −

ρ0c(k)). If λ(k) > 0 for some value of k, the corresponding Fourier component willgrow exponentially in time, such that the homogeneous state is unstable. For λ(k) < 0,small perturbations decay exponentially and the homogeneous state is linearly stable.(The applicability of deterministic DDFT, which assumes local stability, to spinodaldecomposition, which involves local instability, has been doubted by Kawasaki [276,277]. Archer and Evans [14] avoid this problem with the following argument: λ(k) isalways negative for an equilibrium fluid outside the spinodal. A careful treatment isrequired inside the spinodal. In Ref. [14], c(k) is defined in terms of the excess free

88

energy, constructed by a mean-field treatment of attractive interactions. This permitspositive values of λ(k) inside the spinodal.)

This method can be extended to more complicated situations, such as mixtures oftwo particle species [462]. In this case, one needs to linearize two coupled differentialequations, such that finding the dispersion relation becomes an eigenvalue problem.Moreover, in DDFTs for active particles (see Section 4.6.3) that include an orien-tational dependence, a linear stability analysis can be used to test the stability ofan isotropic state against orientational order [410,411]. Mathematical details on linearstability analysis can be found in Ref. [399]. Note that linear stability analysis does notallow to predict the resulting final state. This requires a solution of the full nonlinearproblem, which in most cases is only possible numerically.

Linear stability analysis has found a large number of applications in DDFT. Ex-amples include the determination of the dispersion relation for front-speed calcula-tion [50,462,676,725], traveling waves [345], spinodal decomposition of magnetic fluids[450], spinodal decomposition in a fluid with anisotropic diffusion [726], spinodal andfreezing modes [311,727], the McKean-Vlasov equation [728], the Dean-Kawasaki equa-tion [244], solvent-density modes [388,729], capillary interactions [302], lane formation[323,431], quasicrystal formation [730–733], phase behavior of thin films [457], orienta-tional order of microswimmers [410,411], actively switching particles [461], dynamicsof cancer cells [49], and epidemic outbreaks [50].

7.2.1.2. Front propagation. If a fluid reference state has been found to be unsta-ble against perturbations, it remains to be discussed how the occurring solidificationfront propagates. This can be calculated based on the “marginal stability hypothesis”[734,735], which is applied to DDFT in Refs. [50,462,676,725]. We here present themethod in 1D (see Ref. [462] for a discussion of the 2D case). Suppose we have, bymeans of a linear stability analysis, obtained a dispersion λ(k). If a solidification frontpropagates with velocity v, we can transform the linearized equation of motion forthe perturbations to the co-moving frame and obtain from that the dispersion relationλv(k) = ivk+λ(k). In the co-moving frame, the growth rate at the leading edge of thesolidification front should be zero (since it would leave the frame otherwise). Hence,we get the equations

iv +dλ(k)

dk= 0, (257)

Re(ivk + λ(k)) = 0. (258)

They can be solved for the unknown complex wavenumber k and the unknown velocityv. The selected wavelength is not necessarily that of the resulting equilibrium lattice,which can lead to disorder [462,676].

Front propagation is also studied in PFC models (see Section 6.3) [44,677,736] andAllen-Cahn equations [737]. In Refs. [44,677], front propagation in DDFT is discussedwith an emphasis on how the results compare to PFC calculations.

7.2.2. Separation of time scales

Since standard DDFT is the overdamped limit of more general kinetic theories involv-ing inertia, systematic expansions of such theories in the friction coefficient can beused to obtain DDFT as a limiting case, but also to find inertial corrections. This ispossible using the multiple-time-scale method. The general procedure of deriving over-

89

damped dynamics and corrections from phase-space theories was pioneered by Titulaer[738, 739] and Wilemski [740]. It was applied to DDFT in Refs. [42,134,341,499,500].Moreover, Goddard et al. [398] employed methods of this form for a mathematicallyrigorous derivation of DDFT. Time-scale separation also plays a role in the studyof pattern formation in weakly nonlinear theory [676] and of the violation of thefluctuation-response relation in nonequilibrium steady states [741]. Amplitude equa-tions, the derivation of which is also based on a multiple-scale expansion, are obtainedfor a binary phase field crystal (PFC) model derived from DDFT in Refs. [456,742].

Following Ref. [134], we consider systems with a position-dependent temperatureT (~r). One starts from the nondimensionalized Kramers equation for the phase-spacedistribution Ψ

γndLKΨ(~r,~v, t) =

(∂

∂t+ ~v · ~∇~r + ~F · ~∇~v

)Ψ(~r,~v, t) (259)

with the friction parameter γnd, velocity ~v, force ~F , and Fokker-Planck operator

LK = ~∇~v · (T (~r)~∇~v + ~v). (260)

Equation (259) can be expanded in the eigenfunctions of LK, which are related tothe Hermite polynomials. We replace the physical time scale t by an infinite seriesof auxiliary time scales (τ0, τ1, . . . ) with τn = γ−nnd t and the function Ψ(~r,~v, t) by anauxiliary function Ψa(~r,~v, τn), which depends on the auxiliary time scales that arenow treated as independent. The time derivative becomes

∂

∂t=

∂

∂τ0+

1

γnd

∂

∂τ1+

1

γ2nd

∂

∂τ2+ · · · . (261)

We now expand the auxiliary function in powers of γ−1nd and then each term in eigen-

functions of LK. By inserting all results into Eq. (259) and identifying orders of γ−1nd ,

one obtains a hierarchy of equations for the amplitudes (prefactors arising in the ex-pansion of the auxiliary function in γ−1

nd and eigenfunctions). Compared to simpleperturbation theory, this method avoids secular growth. After some calculation, thelowest nontrivial order gives the usual overdamped DDFT (here with temperaturegradient), while higher orders lead to inertial corrections. Physically, the multiple-time-scale analysis separates the fast time scale on which the momenta relax from thelonger time scales on which the positions relax. The density ρ(~r, t) then “enslaves” allother dynamical variables and follows a closed equation of motion [341,499].

7.2.3. Renormalized perturbation theory

Renormalized perturbation theory has also been applied to DDFT. We discuss ithere following Ref. [213]. The Martin-Siggia-Rose (MSR) procedure [743,744] allowsto write a stochastic differential equation in the form of a field theory based on anaction functional [745], in close analogy to methods used in quantum field theory. Thismethod became a popular tool for the study of the glass transition [746]. A matter ofdebate is whether an ergodic-to-nonergodic (ENE) transition can be found. An ENEtransition, which corresponds to the long-time limit of the density correlation functionbeing nonzero, is predicted by simple MCT [262].

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In particular, a path-integral formalism can be developed for stochastic DDFT.This was done by Kawasaki and Miyazima [116]. We present the method followingRef. [213]. The action S reads

S[ρ, ρ] =

∫d3r

∫dt

(iρ(~r, t)

(∂

∂tρ(~r, t)−Γ~∇·

(ρ~∇ δF

δρ(~r, t)

))−Dρ(~∇ρ(~r, t))2

). (262)

Here, ρ denotes a real auxiliary field and the last term arises from the noise. Theaction (262) is invariant under two types of time-reversal (TR) transformations that,however, are nonlinear. An example is the U transformation

ρ(~r,−t)→ ρ(~r, t), (263)

ρ(~r,−t)→ −ρ(~r, t) +i

kBT

δF

δρ(~r, t). (264)

A loop-expansion of the action (262) leads to problems with the fluctuation-dissipationrelation [747]. The reason for this is that the action (262) can be decomposed into aGaussian and a non-Gaussian part, which are not separately invariant under the TRtransformation. Andreanov et al. [748] solved this problem by introducing two aux-

iliary fields θ and θ. The action then becomes a functional S[ρ, ρ, θ, θ]. Using thenew fields, the U transformation can be linearized. Under the linear transformation,Gaussian and non-Gaussian parts of the new action are separately invariant. Renor-malized perturbation theory can then preserve the fluctuation-dissipation relation ateach order.

Using this idea, Kim and Kawasaki [260, 261, 749] developed a renormalized per-turbation theory from stochastic DDFT. A derivation of MCT at one-loop order byKim and Kawasaki [261] turned out to lack a term that was identified by Kim et al.[213]. While MCT can be recovered by an additional approximation, the full resultof Ref. [213] contains no ENE transition. This conclusion is also supported by thenonperturbative analysis of Bidhoodi and Das [262]. Jacquin et al. [750] studied amodified stochastic model B equation, which, despite having the same equilibriumstate as Eq. (50), showed (even in the mode coupling approximation) no sign of glassybehavior. The case of Brownian particles with momentum was considered by Das [751],who argued that an ENE transition is not supported by the fluctuation-dissipation re-lation also in this case. In Ref. [752], a further TR-symmetry-preserving perturbativescheme was developed.

An introduction to this topic is given in Ref. [233] (here, the results from Ref. [213]are not yet taken into account). Using a Cole-Hopf transformation, a reformulation ofthe action is possible [753]. A diagrammatic derivation of mode coupling results canalso be found in Ref. [754]. Feynman rules and exact results for the case of noninteract-ing particles can be found in Ref. [745]. One can also include the momentum density[755–757] (in Ref. [116], the DDFT action was obtained by eliminating the momen-tum density). Smoluchowski dynamics is considered in Refs. [758–760] and Newtoniandynamics in Refs. [761,762]. In Ref. [763], an MSR integral for polymer dynamics isderived. Orientational degrees of freedom were included in Ref. [239]. Moreover, renor-malization group theory in combination with (hydrodynamic) DDFT is applied to theglass transition of water in Ref. [209].

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7.3. Numerical solutions

Since the DDFT equation (35) is a nonlinear partial differential equation, it is, ingeneral, impossible to solve it analytically. Hence, it is important to have numericalmethods available that can be used to solve it. In general, DDFT has computationaladvantages compared to Brownian dynamics (BD) simulations, since the computa-tional cost does not depend on the density [465]. The computational cost of a BDsimulation increases with the number of particles N , whereas the computational costof a DDFT calculation increases with the desired resolution. As a consequence, for sys-tems with a large number of particles and no small-scale pattern formation, continuumsimulations based on numerically solving a DDFT equation are typically much moreefficient than BD simulations. Compared to dissipative particle dynamics [764–768],which is another method for simulating fluids on the mesoscale, DDFT is applica-ble down to microscopic scales [769]. In Ref. [770] (see also Refs. [771,772] for earlierwork), a hybrid particle-continuum simulation method is presented, where the level ofresolution switches adaptively between BD and DDFT field propagation.

A significant amount of work has been done in numerical mathematics on the solu-tion of partial differential equations of the gradient-dynamics type (see Section 6.2),which DDFT belongs to. However, only a few of these methods are specifically de-signed for DDFT. Many of them address PFC models [773–782], the Allen-Cahn andCahn-Hilliard equations [783–789], the Swift-Hohenberg equation [790], or the Poisson-Nernst-Planck (PNP) equations [791]. For PFC models, Backofen et al. [674] presenteda semi-implicit integration scheme in order to avoid the problems of explicit (time-steprestrictions) and implicit (nonlinear equations) methods. Due to the close mathemat-ical relation of these models to DDFT, it is likely that similar approaches can also beuseful here.

Moreover, DDFT can benefit from methods developed for static DFT. Some ofthese methods have already been applied to DDFT (such as numerical continuation),whereas others have potential for future applications (such as machine learning). Avariety of numerical strategies for DFT are discussed in Ref. [792], including real-spacenumerics, fast Fourier transformations, parallelization, preconditioning, and pseudo-arc-length continuation. Numerical methods for polymer DFT can be found in Ref.[793]. More recently, machine-learning methods have been used to approximate classi-cal density functionals in DFT [794]. Machine learning is already successfully appliedin quantum DFT (see Refs. [795,796] for reviews).

A variety of authors have discussed the discretization and integration of DDFT, fo-cusing on a variety of specific problems. For stochastic DDFT, a finite-volume methodwas developed by Russo et al. [797]. A simple finite-difference integration scheme fordeterministic DDFT can be found in Ref. [460]. Discretization of deterministic DDFTis discussed in Ref. [203]. Differential equations arising in DDFT can be stiff for finegrids. Since implicit Runge-Kutta algorithms, which are more efficient than simpletime-stepping methods in such situations, cause difficulties due to the large dimen-sionality and nonlinearity of the problem, specialized explicit Runge-Kutta methodsare employed [335]. A discussion of the finite-element discretization of DDFT, with anemphasis on the relation of stochastic and deterministic theories, the introduction ofthermal fluctuations, and the relation to the physics of coarse graining, can be foundin Ref. [278]. In Ref. [798], finite-volume schemes for general transport equations thatinclude DDFT as a limiting case are presented.

Nonlocal terms resulting from the excess free energy are particularly difficult to treatnumerically. Convolution terms on Cartesian grids can be evaluated efficiently using

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fast-Fourier-transformation methods. However, these are difficult to apply on generalmeshes [422]. Since hard spherical disks cause geometrical difficulties when using thestandard method of uniform Cartesian grids, partially refined grids are useful [335].A spectral method for both DFT and DDFT is presented in Ref. [206]. It involveschoosing a discretization scheme where the mesh is dense close to walls, where largerdensity variations are expected. A Clenshaw-Curtis quadrature is used to evaluatethe DFT convolution integrals. Nold et al. [161] developed a pseudo-spectral methodfor the evaluation of the nonlocal terms. This scheme is discussed for a variety ofcontexts in DFT and FMT and also applied to DDFT where it is checked for massconservation. On the surface of a sphere, the convolutions can be efficiently computedusing expansions in spherical harmonics [454].

For DDFT calculations in higher dimensions, which can become rather slow due tothe large number of lattice points required, parallelization is important. The paral-lelization of DFT calculations is discussed in Refs. [799,800]. Fraaije and Evers [801]presented parallelized DDFT algorithms in FORTRAN.

Another useful method in DDFT is numerical continuation [802–808], which allowsto track solutions of an equation as a function of a control parameter. Numerical con-tinuation is discussed in the context of DDFT in Refs. [370,543,809]. It was used byGomes et al. [491] to determine bifurcation diagrams for the McKean-Vlasov equa-tion (a DDFT-type equation). Pototsky et al. [727] employed continuation methodsto calculate bifurcations of clusters. In Ref. [206], an algorithm for the numerical con-tinuation of DFT is presented.

DDFT (in the form discussed in Section 3.2.3) also forms the basis of the MesoDynsoftware [57,810], which can be used to simulate polymer materials. The time scalesinvolved in phase separation and pattern formation of block copolymers are typicallytoo large to be accessible by molecular dynamics simulations [810], in particular forindustrial applications in soft nanoscience and material design. On the other hand,the time scales in industrial processing are much shorter than those required for fullrelaxation to thermodynamic equilibrium, such that the nonequilibrium behavior isimportant [811]. DDFT-based methods allow for a simulation of the time-dependentthermodynamic behavior on the mesoscale [810]. A discussion of various multiscalesimulation methods for materials can be found in Refs. [812,813].

For kinetic extensions of DDFT (see Section 4.7.3), the use of lattice Boltzmannmethods [814–820] has been suggested by Marini Bettolo Marconi and Melchionna[477, 505, 506]. First, the phase-space distribution function and the collision opera-tor are expanded in Hermite polynomials. Second, the moments are evaluated usingGauss-Hermite quadratures. Third, the distribution function is propagated using aforward Euler procedure. The lattice Boltzmann method has certain advantages overthe solution of the coupled hydrodynamic equations, such as an easier handling ofboundary conditions [506]. Baskaran et al. [519] developed implicit finite-differencemethods for a DDFT with momentum density that takes the form of a compressibleNavier-Stokes equation.

A numerical iterative scheme allowing to determine the external force field thatgenerates a certain density and velocity profile can be found in Ref. [30]. The existenceof a mapping between motion and force field is implied by power functional theory(PFT) (see Section 5.2). In the context of PFT, numerical techniques have moreoverbeen developed to sample the one-body current from BD simulations, where one hasto take the stochastic nature of the underlying dynamics into account [30,32].

Finally, Monte Carlo (MC) simulations are often combined with DDFT. We onlydiscuss a few examples here. The stochastic DDFT (40) by Kawasaki (see Section 3.2.2)

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can be solved by mapping it onto a kinetic lattice gas model. This model can then besolved using ordinary MC techniques [227,232,821]. Dynamic MC simulations can beemployed to test approximations made in DDFT [822]. The scheme for the determi-nation of superadiabatic forces presented in Ref. [287] requires an adiabatic referencepotential, which can be obtained from MC simulations [287,340]. In Ref. [397], station-ary density profiles obtained from a DDFT for flow-driven hard spheres are comparedto MC results. Moreover, DDFT can help to incorporate hydrodynamic effects intoMC simulations [404].

8. Applications

Having discussed how to analyze a DDFT equation, we finally turn to applications ofDDFT. This is a very broad and diverse field, which we present here ordered by sys-tems DDFT is applied to (Section 8.1) and by phenomena that are described by DDFT(Section 8.2). In Section 8.1, we start with the systems introduced in Section 3, namelycolloidal fluids (Section 8.1.1), atomic and molecular fluids (Section 8.1.2), and poly-mers (Section 8.1.3), for which standard DDFT has been developed. We then addressthin films (Section 8.1.4) and glassy systems (Section 8.1.5), which are composed ofthe same materials, but have special properties that affect their description in DDFT.Different materials come into play when considering granular media (Section 8.1.6)and plasmas (Section 8.1.7). Driven (Section 8.1.8) and active (Section 8.1.9) softmatter are also nonstandard cases, since in these systems typical close-to-equilibriumapproximations of DDFT tend to break down. Finally, we present electrochemical (Sec-tion 8.1.10) and biological (Section 8.1.11) systems, which are particularly complex.For phenomena (Section 8.2), we again start with the most basic aspects, which arerelaxation (Section 8.2.1) and phase separation (Section 8.2.2). This naturally leadsto pattern formation (Section 8.2.3), which can be a consequence of phase separation,and nucleation and solidification (Section 8.2.4), which is a form of relaxation (to aparticular thermodynamic equilibrium state) and can also lead to pattern formation.We then turn to specific applications where pattern formation is one (though not theonly one) relevant effect, namely chemical reactions (Section 8.2.5), disease spreading(Section 8.2.6), and feedback control (Section 8.2.7). Finally, we discuss sound waves(Section 8.2.8). A further important aspect are “theoretical applications”, which in-volve the derivation of new dynamic equations from DDFT. Important examples in-clude MCT and PFC models, which have both been explained in previous sections(Sections 6.1 and 6.3, respectively), and the dynamics of the van Hove function, whichis discussed in Section 8.3.

8.1. Systems

8.1.1. Colloidal fluids

Colloidal fluids are a standard system of application for DDFT. This case is discussedextensively in Section 3, in particular in Section 3.5.3.

An example is presented in Fig. 7 on the following page, which is adapted from Ref.[41] and shows the time evolution of the density profile of colloidal spheres in a peri-odically switching optical trap. The steady state is a driven breathing mode. Resultsfrom BD simulations are also shown. While there are small deviations resulting fromsuperadiabatic contributions, the general agreement between theory and simulation is

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Figure 7. Colloidal spheres with diameter σ in a periodically switching optical trap. Density profiles ρ(r, t)

obtained by DDFT and BD simulations are shown at different times t, where τB = σ2/D denotes the Browniantime. Adapted from Fig. 2 from Ref. [41].

very good.

8.1.2. Atomic and molecular fluids

Atomic and molecular fluids are another standard system of application for DDFTand discussed in Sections 3 and 4.7.1.

8.1.3. Polymers

Polymers are the third standard system of application introduced in Section 3. Theywere among the first systems studied using DDFT and the theory presented in Sec-tion 3.2.3 is now one of the most widely used methods in the simulation of polymersystems. Applications include phase behavior [55,360–362,433,524,830–865], systemsunder shear [51,433–445], pattern formation and self-organization [56,269,270,866–881], effects of electric fields [882–885], adsorption/desorption kinetics [886], thin films[55,56,269,362,843,859,861,887], mechanical properties [446,888–890], and the responseof materials to embedded nanoparticles [437,891].

DDFT has gained importance in industrial applications due to its applicability asa simulation technique. In particular, the MesoDyn method [57,810,825] is widelyused for polymer simulation. Moreover, DDFT can be combined with BD in hybridparticle-field simulations [770,826] (see also Refs. [771,772,827,828]). Some commentson parallelization can be found in Ref. [801]. Various alternative approaches are dis-cussed in Refs. [763,793,903–906]. For general reviews of (co)polymer modeling andsimulation, see Refs. [436,825,885,892–902]. A fairly complete overview over the liter-ature on polymer DDFT is given in Table 2.

Since polymer DDFT is a large topic on its own that has been reviewed elsewhere[269,270], we only discuss a few studies here as examples. All of them consider polymerthin films, which have increasing importance for nanotechnology. Knoll et al. [362]used DDFT for computer simulations of copolymer thin films. The phase diagram wasobtained both theoretically and experimentally. Wetting layers and lamella formationwere observed. Knoll et al. [55] compared simulation data from DDFT to experimentson nanostructured fluids. These also show a lamella phase, as do the thin films of ABCtriblock copolymer, whose self-assembly was studied theoretically and experimentallyby Ludwigs et al. [56]. All simulations reported here employ the MesoDyn software.

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Topic ReferencesGeneral theory [263–267,823,824]Original derivations [10,124]Extensions:• External potential dynamics [268]• Compressibility [126]• Viscoelasticity [525]Software:• MesoDyn [57,810,825]• Hybrid BD-DDFT simulations [770–772,826–828]• Other aspects [801,829]Applications:• Phase behavior [55,360–362,433,524,830–865]• Sheared systems [51,433–445]• Pattern formation [56,269,270,866–881]• Electric fields [882–885]• Adsorption/desorption [886]• Thin films [55,56,269,362,843,859,861,887]• Mechanical properties [446,888–890]• Response to nanoparticles [437,891]Reviews:• Polymer DDFT [269,270]• Polymer simulation [436,825,885,892–902]

Table 2. Overview over the literature on polymer DDFT.

Other applications of DDFT to polymers [132,907] are based on the methods pre-sented in Sections 3.2.1 and 3.2.2. Depending on the effects one is interested in, thepolymers can be assumed to be interacting or noninteracting [18]. Applications includediffusion in colloid-polymer mixtures [203], encapsulation kinetics of hollow hydro-gels consisting of polymeric shells [908], and polymer conformation [222,223]. Polymerchains are described using a DDFT with hydrodynamics in Refs. [210,403]. Cahn-Hilliard-type models can also be applied to polymer systems [909]. Further applicationsof DDFT include thin films (see Section 8.1.4) and biology (see Section 8.1.11).

8.1.4. Thin films

While they also fall into the category “fluids and polymers” discussed in Sections 8.1.1to 8.1.3, thin films deserve a separate discussion due to their theoretical and practicalimportance. Polymer thin films have already been addressed in Section 8.1.3. Here,we give a broader overview. The wetting behavior of droplets and thin films is oftenmodeled through phenomenological equations based on long-wave hydrodynamics orgradient dynamics (see Refs. [648,910–912] for reviews), which exist in deterministicand stochastic forms [913]. DDFT provides an alternative based on microscopic particledynamics (see Ref. [458] for a review). It allows for a description of various impor-tant effects, such as contact angle hysteresis, capillary-driven motion [507], wetting ofconfined colloidal fluids [272], and nanodroplet coalescence [198]. Interesting insightsinto wetting behavior can already be gained using static DFT [914–923]. Numericalaspects of wetting in (D)DFT are discussed in Ref. [206].

Frequently, one studies solvents in which nanoparticles are immersed. Dewetting

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processes are more complex in such systems, since various transport and evaporationprocesses are involved [459]. As in the stationary case [921], the system is often modeledas a lattice. Consequently, DMFT (see Section 6.5) is a method used in this field[701,712,716,718]. The sites of the lattice can be unoccupied, occupied by liquid, oroccupied by a nanoparticle [457]. The free energy governing the DDFT can be obtainedby a mapping from the lattice Hamiltonian or through gradient expansions of the freeenergy functionals for continuous systems [458]. The mobility of the nanoparticlesdepends on the density of the solvent, since they are transported through the film andcannot move on the dry substrate. Moreover, due to evaporation processes governedby a difference between the chemical potential of the liquid on the surface and thatin a reservoir, the DDFT equation gets an additional nonconserved term that has anAllen-Cahn-type form [457]. Denoting the densities of liquid and nanoparticles by ρl

and ρn, respectively, this gives

∂

∂tρn(~r, t) = ~∇ ·

(Mn(ρn, ρl)~∇

δF [ρn, ρl]

δρn(~r, t)

), (265)

∂

∂tρl(~r, t) = M c

l~∇ ·(ρl(~r, t)

δF [ρn, ρl]

δρl(~r, t)

)−Mn

l

δF [ρn, ρl]

δρl(~r, t), (266)

where the mobility Mn of the nanoparticles depends on the local densities, whereas themobilities M c

l and Mnl for the conserved and nonconserved part of the liquid motion,

respectively, are constant. DDFT models allow to analyze a variety of effects in thecontext of dewetting and evaporation, such as spinodal decomposition, nucleation,and fingering [457–459]. Moreover, a thermodynamic equivalent of the coffee-ring staineffect can be found [460]. A review is given by Ref. [924]. Simulations and experimentsare described in Ref. [925].

Standard DDFT is also applied to study drying mixtures of colloids and polymers.In particular, one is here interested in stratification. It is found, in agreement withmolecular dynamics simulations, that initially mixed fluids stratify with smaller col-loids on top [188], while in colloid-polymer mixtures the result depends on the polymerlength and radius of gyration [339]. As a mechanism for stratification in drying films,diffusiophoresis has been proposed [926]. Another application of DDFT in the contextof thin films is the dynamics of surfactants [927,928]. Their density can be describedby a diffusion-advection-reaction equation, where the diffusive current of DDFT iscombined with an advection term from the Marangoni flux and reaction terms forad-/desorption and photoisomerization [419]. Ye et al. [208] studied the oleophobicityof polymer films. Thin films have also been considered in the context of the polymerDDFT presented in Section 3.2.3 [128,843,859,861,887], some examples are discussedin Section 8.1.3. Finally, DDFT can also be applied to wetting and drying in activesystems [23].

8.1.5. Glassy systems

When entering a glassy state, soft matter systems exhibit a variety of interesting non-standard properties. Like thin films (see Section 8.1.4), glasses are therefore discussedseparately. However, the application of DDFT to glasses is more controversial andtherefore requires a particularly careful treatment.

The dynamics of supercooled liquids and glasses is often modeled using MCT, whoseformalism and relation to DDFT are discussed in detail in Section 6.1. It models glassybehavior as an ergodic-to-nonergodic (ENE) transition. The applicability of DDFT in

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this field is a question that is discussed very controversially. On the one hand, stochasticDDFT was specifically designed as a generalization of MCT for the treatment ofsupercooled liquids [114,229,230,929], DDFT was used to investigate the origin ofnonergodicity in MCT [930], MCT was derived from DDFT [114,131], and DDFTreproduces glass-like relaxation behavior [342]. On the other hand, it was argued thatDDFT cannot describe caging and nonergodicity effects relevant for glasses [39,285],renormalized perturbation theory showed no ENE transition [213], and the applicationto glasses seems incompatible with the fact that the derivation of DDFT involvesneglecting memory effects [40].

To resolve these issues, it is important to distinguish between three different ques-tions:

(1) Can DDFT reproduce MCT?(2) Can DDFT describe the physics of (real) glasses?(3) Can DDFT describe nonergodic systems?

The subtle relation of DDFT and MCT is discussed in Section 6.1. Stochastic DDFTwas developed in the context of MCT by Kirkpatrick and Wolynes [95] and Kawasaki[114]. Based on stochastic DDFT, one can derive an equation of motion for the correla-tor of the density fluctuations that has the form of an MCT equation. This correspondsto the microscopic MCT if the DDFT describes the microscopic density, whereas acoarse-grained stochastic DDFT leads to an MCT-like equation for the fluctuations ofthe coarse-grained density. A derivation of the latter type [131] is presented in Sec-tion 6.1. Other ways of deriving MCT include projection operators [114] and renor-malized perturbation theory [213] (see Section 7.2.3). Deterministic DDFT and MCTare complementary approaches that correspond to making different approximations tothe full microscopic dynamics [24,40].

The second issue was the original motivation for stochastic DDFT: The standarddescription of cage effects in simple MCT as an ENE transition does not apply toreal materials due to hopping effects resulting from thermal fluctuations (although thedeviations from MCT due to thermal fluctuations are comparatively small in dense col-loidal liquids, since colloids are very large). These fluctuations are included in stochas-tic DDFT [234]. In dense liquids, momentum and energy can be transferred on timescales much shorter than those on which the particle configuration can change, suchthat it is natural to use the density as the only slow variable in such a theory. Fuchizakiand Kawasaki [226] showed that stochastic DDFT is capable of describing mode cou-pling mechanisms. Stochastic DDFT gives access to the later stages of freezing in asupercooled liquid, where the system relaxes from the glassy minima by thermallyactivated hopping [227]. For practical purposes, the description can be simplified byusing a lattice model [226,227]. Another important method in this context is renor-malized perturbation theory (see Section 7.2.3). A brief overview over both MCT andstochastic DDFT is given by Ref. [235]. Further discussions can be found in Refs.[233,931].

With deterministic DDFT, a description of dense atomic liquids is also possible[131]. A specific problem is, however, that if a system gets trapped in a local minimum,deterministic DDFT will predict that it is trapped forever [15]. A further problemregarding dense colloidal suspensions is that hydrodynamic interactions are impor-tant here, which affects both deterministic and stochastic methods [234]. Within thederivation of DDFT from the projection operator method by Espanol and Lowen [16](see Section 3.3.4), this is reflected by the fact that one obtains the general nonlocalequation (105), which can be approximated by the local form (35) if one assumes

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a dilute colloidal suspension. At finite colloidal densities, hydrodynamic interactionsbecome important and they are naturally included in extensions of standard DDFTthat are based on the projection operator formalism, such as EDDFT [40] (see Para-graph 5.3.2.1).

DDFT has been applied to glassy behavior in a variety of contexts: DDFT can beused to calculate friction forces in supercooled liquids [644,645]. The density of theglass transition was estimated by a calculation of the time required for crossing anenergy barrier by Archer et al. [343]. Berry and Grant [501] employed a stochasticDDFT with inertia to model the glass transition, showing agreement with MCT andindications of fragile behavior. Disorder can be found to arise from the wavelengthsselected by front propagation [676] (see Paragraph 7.2.1.2).

The main problem, however, arises from the third question: As shown by Reinhardtand Brader [285], DDFT allows hard particles to pass through each other, which makesits application to nonergodic states problematic. This was studied in detail for a 1Dsystem by Schindler et al. [39]. A system of hard rods in 1D is nonergodic because it hasa strict particle order, i.e., if particle 1 is initially on the left of particle 2, it always willbe there. However, when labeling the particles and describing their dynamics using anadiabatically exact canonical DDFT for mixtures, one finds that, in contrast to the BDdescription, the particle order is not conserved. Schindler et al. [39] inferred that cagingeffects as arising in glasses are not incorporated in standard DDFT. As a solution,they suggested the use of an asymmetric interaction potential. Based on this idea,Wittmann et al. [273] derived a statistical-mechanical theory for ordered ensembles.This forms the basis of order-preserving dynamics (OPD), a modified DDFT whichtakes the conservation of particle order into account.

For stochastic DDFT, in particular in the formalism by Dean [105], the existenceof an ENE transition is a widely debated and investigated topic, in particular inthe context of the path integral formalism (see Section 7.2.3). It was discussed bothwhether such a transition exists at one-loop order [759] and whether higher-order loopscan affect the result [261]. Results by Kim et al. [213] indicate that such a transitiondoes not exist, even though MCT can be recovered by further approximations.

What is in need of explanation is then the fact that DDFT nevertheless some-times gives useful results when applied to glasses and MCT. Two aspects play a rolehere. First, approximations can make an important difference. As already pointed outby Marini Bettolo Marconi and Tarazona [12], the exact free energy of deterministicDDFT has a global minimum at equilibrium (corresponding to DFT), but approximatefree energy functionals can have local minima. These local minima, in which the de-terministic dynamics can get trapped forever if an approximate free energy functionalis used, correspond to very long but finite relaxation times in the exact description.Such minima have been found by Archer et al. [343] and were used to estimate tran-sitions to nonergodic regimes resulting from the system being unable to escape fromthese minima. In treatments with more accurate free energy functionals, these min-ima do no longer arise [159,293]. The dynamic arrest observed in Refs. [159,342] is nolonger present if more accurate FMT-type free energies are used, such that Stopperet al. [159, 293] attributed it to a too strongly simplified free energy functional. Instochastic DDFT, the treatment in renormalized perturbation theory by Kim et al.[213] (see Section 7.2.3) showed no ENE transition. Nevertheless, making additionalapproximations allows to recover MCT. A second point is that a lack of nonergodicityin DDFT only makes it inapplicable to glasses insofar they are actually nonergodic. Asindicated above, the argument behind the application of stochastic DDFT to glasses isprecisely the fact that, in the long run, thermal fluctuations will lead to a restoration

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of ergodicity and to the approach to an equilibrium state. From this perspective, theabsence of a nonergodic transition is an advantage rather than a problem.

Ideas from DDFT have also been employed in the study of activated hopping inglassy liquids. This is done in the theory by Schweizer [932], which he derived build-ing up on earlier work in Refs. [933,934]. The idea is to use the single-particle radialdisplacement r(t) as a slow dynamical variable. At short times, the particle movesdiffusively. A caging force −∂Feff/∂r, derived from an effective free energy Feff thatis based on ideas from DFT, but has no rigorous equilibrium meaning, captures thetendency to localize at higher volume fractions which is described by MCT [932].Finally, thermal noise leads to ergodicity-restoring activated hopping. A more system-atic derivation of the resulting dynamic equation (267) proceeds as follows: Startingfrom the single-particle Langevin equations, a formally exact stochastic theory for thesingle-particle density operator is obtained. Multiplying by ~r2 (motivated by using thelocal displacement as an order parameter), integrating, and averaging gives a dynamicequation for the squared displacement of a tagged particle r2(t). This equation involvesthe two-body correlation, which is then related to the functional derivative of a freeenergy as in the standard DDFT approximation. After using the chain rule, one finds

ξsdr(t)

dt= −∂Feff(r(t))

∂r+ χ(t) (267)

with the short-time friction constant ξs, the effective free energy

Feff(r(t)) = −3kBT ln(r(t))− kBT

2

∫d3k ρ

c(k)2

(2π)3S(k)e−

1

3k2r(t)2 , (268)

where ρ is a constant number density, c(k) is the Fourier-transformed direct pair-correlation function, and S(k) is the dimensionless collective structure factor, andthe zero-mean delta-correlated noise χ. This method can be used to describe hopping,diffusion, and relaxation in glassy fluids [935–951]. Reviews are given by Refs. [952,953].

Aging effects relevant for the glass transition can also be described by nonequilib-rium generalized Langevin equations (see Section 6.4) that include DDFT as a specialcase [692,695,696]. Brader and Schmidt [24] derived from an excess dissipation func-tional a generalized theory including both deterministic DDFT and MCT as limitingcases. The relation of memory functions to power functionals is further explored inRef. [31]. Glass-like relaxation in gels can be observed in DMFT [721].

8.1.6. Granular media

We now turn to systems composed of different materials, starting with granular me-dia (such as sand or powder). Granular gases consist of particles exhibiting inelasticcollisions. A dynamical theory for a thermostatted one-dimensional granular fluid isderived in Ref. [341] using multiple-time-scale methods (see Section 7.2.2). It has theform of a DDFT equation with additional terms from inelastic collisions. The discus-sion is extended in Ref. [500]. More recently, a DDFT for granular fluids was derivedby Goddard et al. [186], which couples dynamic equations for mass, momentum, andenergy density. The dissipative effects of inelastic collisions are accounted for by mo-ments of a collision operator.

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8.1.7. Plasmas

Plasmas are, despite having physical properties very different from those of colloidalfluids, also studied using methods from the field of DDFT. The hydrodynamics ofplasmas can be described using the one-particle density defined on phase space, whichis governed by a dynamic equation that depends on the two-particle density (which,in turn, depends on the three-particle density and so on). A partial closure of thisBogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy can be obtained fromDDFT, which relates the two-particle density to the free energy functional. Com-bined with relaxation closures for stress tensor and heat flow, this allows to deriveclosed hydrodynamic equations for plasmas [327]. When extended to mixtures, suchapproaches allow to model astrophysical plasmas in white dwarfs and neutron stars[328]. Moreover, these approaches can also be extended to quantum mechanics [47](see Section 4.11).

8.1.8. Driven soft matter

Systems can deviate from those considered in “standard” applications of DDFT notonly because they are composed of different materials, but also because they are drivenaway from equilibrium. This can be a consequence of external drives or of activity (thelatter case is discussed in Section 8.1.9). Here, we consider driven soft matter, which isa field of application for both DDFT and extensions such as PFT [331]. This includes,in particular, systems that are subject to a time-dependent external potential [17].A review of instabilities in driven soft matter (including DDFT applications) can befound in Ref. [326]. More general aspects of colloids in external fields are reviewed inRef. [954]. Examples of DDFT studies of driven soft matter include colloids that aredragged through a polymer solution at a constant rate [18], colloids in time-dependentoptical traps exhibiting breathing modes [41], attractive particles that are subject toan ac or dc drive [310,311], flow through constrictions [314], ions in narrow confinement[53], and flow-driven particles [397,430]. Moreover, driven nonequilibrium systems havebeen mapped to equilibrium systems that can then be studied using equilibrium DFT[955]. One can also study stochastic driving, such as by a thermostat [341].

Early studies were interested in potentials of the form U1(x−vt) that move at a con-stant rate, a scenario that has been studied for flow in narrow channels [724], draggedparticles [18], and sheared soft colloids [19]. As discussed in Section 7.1, steady currentscan arise as solutions of the DDFT equation in this scenario [724]. Additional effectscan be found if the inertia of the particles is taken into account [291]. Comparisonswith BD simulations show that DDFT can also be applied to more complex time-dependent potentials such as oscillating cavities [346]. A further extension are drivensystems with orientational degrees of freedom [176]. Synchronized driven dipoles havealso been described in an equilibrium DFT [956].

DDFT has also been used to model pattern formation in driven soft matter. Anexample is nonequilibrium lane formation [323,431]. Laning has been incorporated inDDFT by adding currents corresponding to the external drive [324,325]. It can alsoarise as a consequence of shear [429]. In Ref. [332], laning is studied using PFT. Periodicexternal fields can lead to band formation [309]. Moreover, time-dependent potentialsare a way to design spatiotemporal patterns in driven colloidal systems [957]. Instabil-ities resulting in time-dependent density profiles are a possible consequence of externaldrives [311]. Driven tracer particles in colloidal systems have been observed to inducelocal phases such as colloid-poor bridges and cavitation bubbles [432]. New types ofdynamical instabilities have been found to arise as a consequence of time-dependent

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external fields combined with intrinsic phase separation [449]. Finally, DDFT has beenapplied to synchronization and dynamic mode-locking in colloidal particles that aresubject to a modulated driving force [313].

A further application of DDFT is the study of depletion forces in driven systems[543]. Depletion forces in nonequilibrium systems can be shown to have very differentproperties than their equilibrium counterparts [958]. An interesting consequence ofdepletion forces is the “Brazil nut effect”, where heavy particles float on top of lighterparticles [294].

There are, however, certain difficulties in applying DDFT to driven systems, sincethe presence of external drives can distort the correlation functions from their equilib-rium form [289]. Improved theories have also been derived [286]. Lips et al. [308] foundresults of DDFT to agree with those small-driving approximations, which (like DDFT)underestimate mean interaction forces in systems driven by drag forces. Moreover, sim-ple advected DDFT equations cannot capture effects of shear flow, since for symmetryreasons they are solved by the unsheared density field. Modifications are required tocapture effects of shear flow [137,182,429] (see Section 4.4.2 for a discussion).

8.1.9. Active soft matter

DDFT for active particles (see Section 4.6.3) is used to describe a variety of effectsin active soft matter. Many studies are interested in confined systems, such as self-propelled rods in a confining channel [20], microswimmers confined to a plane [22],circle swimmers in a circular trap [409], and ABPs in a 2D trap [298]. A typical effectis the accumulation at boundaries and surfaces [23], e.g., in the form of transientclusters [20]. Active DDFTs based on an “effective” description resembling a passivesystem allow to investigate steady states [23,205] and in particular to apply conceptsof pressure [205] and chemical potential [299] from equilibrium thermodynamics in theactive case. Moreover, DDFT allows to describe the spatial dependence of the density[297] and is thus suitable for describing pattern formation in active matter [461] (seealso Section 8.2.3). For example, Arold and Schmiedeberg [498] considered laning inunderdamped active systems. The DDFT for microswimmers has been applied tostudy stability against orientational order and collective motion as well as behaviorof pushers and pullers [410,411], localizing effects in circle swimmers [409], and theformation of “fluid pump states” [22]. Reaction DDFT can model the phase behaviorof actively switching soft colloids [418]. Menzel et al. [493] used active DDFT to derivean active PFC model. Finally, active PFT is used to find steady-state sum rules [296]and nonequilibrium phase behavior [32,300].

8.1.10. Electrochemical systems

DDFT has also become an important method in electrochemistry, e.g., in the descrip-tion of electrolyte conductivity [959] or charging processes in capacitors [475,960], asit allows for a microscopic description of ion interactions [961]. In this field, two linesof work can be distinguished. The first one is by Chandra, Bagchi, and coworkers, whoapplied DDFT to obtain dynamic equations for ions [117,962] and to study viscosity,friction, and conductivity in electrolyte solutions [118–120,406,641,643,963–966] (seeRef. [967] for a review). Here, a combination of DDFT and MCT is used. Ion conduc-tance is driven by ion diffusion, which is related to ion friction. MCT can be used to

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calculate ion friction and self-diffusion coefficients. DDFT provides the force [117]

~Fs(~r, t) = kBTρs(~r, t)~∇Ns∑i=1

∫d3r′ c(2)

si (~r, ~r′)∆ρi(~r′, t) (269)

with the density of the tagged ion ρs, number of ion species Ns, direct pair-correlation

function c(2)si corresponding to the correlation between species i and the tagged ion,

and density fluctuations ∆ρi of ion species i. The friction δξs is then obtained fromthe Kirkwood formula [117,119] (see also Refs. [968,969])

δξs =1

3kBT

∫d3r 〈~Fs(~r, t) · ~Fs(~r, 0)〉. (270)

DDFT is also used here to calculate van Hove functions [117] (see Section 8.3).The second line of work concerns the Poisson-Nernst-Planck (PNP) equations, which

are a very popular model for ion transport. They are given by [185]

∂

∂tρi(~r, t) = Γ~∇ ·

(kBT ~∇ρi(~r, t) + ρi(~r, t)(Qi~∇ψ(~r, t) + ~∇U1(~r))

), (271)

~∇2ψ(~r, t) = − 1

ε0εr

Ns∑i=1

Qiρi(~r, t) (272)

and describe the density ρi of ion species i as well as the electrostatic potential ψ,where Qi is the charge of an ion of species i and εr is the dielectric constant of thesolvent. These equations describe diffusion of ions in the presence of an electrostaticpotential ψ, which is determined by the configuration of the ions via the Poissonequation. A microscopic derivation can be found in Ref. [970]. The dynamic equationfor the ions can be obtained from DDFT by including an electrostatic contributionin the free energy [354,414,415,479,483]. In Refs. [477,481,482], the PNP equationsare obtained from multicomponent kinetic theory. The PNP equations can also becoupled to a flow field ~v(~r, t), which gives the Poisson-Nernst-Planck-Navier-Stokesequations [414,415,971]. A further hydrodynamic theory for complex charged fluidscan be found in Ref. [478]. However, the standard PNP equations do not includeexcluded-volume effects and are therefore valid only in the dilute limit [791,972–975].A phenomenological solution to this problem was proposed by Bazant et al. [976].

To give a more accurate description, DDFT is used to model the ions as chargedhard spheres [476]. The resulting theory is an extension of the PNP equations that in-cludes a hard-sphere contribution in the excess free energy, which is typically obtainedfrom FMT [184,185]. Dispersion forces can also be incorporated, leading to a nonmono-tonic time evolution of the surface charge of electric double layers [192]. In addition,excluded-volume effects can be shown to affect the dielectric constant [503]. Impedanceresonance can be studied by analyzing the effects of a time-dependent external electricfield on the ions in DDFT [53]. Further applications include reversible heating [484],viscosity effects [185], ion dynamics in nanopores [480,485,977], and colloid-interfaceinteractions [354]. Lee et al. [978] mentioned the assumption that the ion density islow compared to the solvent density as a limitation of DDFT. Finally, the dynamicsof the surface charge density field in an ion system can, by introducing an effectivepotential, be written as a (stochastic) DDFT [979]. Stochastic DDFT has also beenapplied to fluctuations of counterions near a charged plate [246]. In Refs. [414,415],

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a PNP model for the surface charge density involving source terms is derived usingDDFT. Effects of kinetic dielectric decrement were studied in Ref. [349]. A remainingchallenge is the accurate prediction of relaxation times [980]. Applications of DFT tocapacitive systems are reviewed in Ref. [981] and the theory of solid electrolytes isreviewed in Ref. [982] (both reviews include some remarks on DDFT).

The PNP equations are also used in biology to model ion channels (see Refs. [983–985] for an overview). To account for interactions of the ions, the PNP equations canbe supplemented by a chemical potential that is derived from a DFT free energy [986](see also Ref. [987] for the DFT of charged hard spheres), giving a DDFT-like equation.The PNP equations extended in this way then provide a model of calcium channels,which are relevant for muscle contraction [351,352]. Similar methods are applied instudying transport through membranes [988] and nanopores [989].

8.1.11. Biological systems

Biological systems, which are particularly complex, combine many of the topics pre-sented so far: Biological microswimmers are a prototypical example of active matter[22], transport through ion channels is a case of electrochemical transport [351], poly-mers are important in biology [886], and biofilms can be described using thin-filmmodels [990]. A good example for a biological application of DDFT is the study ofproteins [375], in particular protein adsorption. Here, one analyzes how proteins ad-sorb on a surface where polymers are chemically attached. This is important, e.g., inthe development of biomimetic materials, where one wishes to avoid adsorption onartificial organs [991], or for drug delivery, where adsorption of the drug carriers in theblood stream needs to be delayed so that they can reach their target [48]. Moreover,protein adsorption is central in various fields of biotechnology [356]. Experimental re-sults on protein adsorption (with comparison to theory) can be found in Ref. [992].One is interested in both minimizing the equilibrium adsorption and in the control ofthe adsorption kinetics.

With DDFT, it is possible to obtain construction guidelines for the prevention ofadsorption [48]. In the simplest case, one assumes the solvent and the polymers to relaxvery quickly and uses a DDFT equation to model the dynamics of the protein density.The free energy, whose functional derivative gives the chemical potential determiningthe protein adsorption, depends on the configuration of polymers and solvent [355].More complex approaches include various types of proteins and allow the proteins to bein various configurations. Intermolecular repulsion can be taken into account throughan additional volume constraint [463]. Extended models allow to incorporate effects ofelectrostatic interactions and salt concentrations on the adsorption behavior [464]. Thefree energy of protein adsorption models includes, in addition to the ideal gas term andthe external potential, the adsorption free energy for protein-specific protein-surfaceinteractions, a term for electrostatic interactions, and the usual excess free energyfor protein-protein interactions [356]. Mixtures of various protein types can also bestudied [463,464]. An interesting application of DDFT is the Vroman effect [993],where proteins show a nonmonotonic adsorption behavior (typically, smaller proteinsadsorb first and are then replaced by larger ones) [463,474]. DDFT-based predictionsagree with experiments much better than predictions from simple diffusion equations[356]. Reviews are given by Refs. [333,994–999].

A further biological application is the study of cancer growth. A DDFT model forcancer has been developed by Chauviere et al. [46] and further explored by Al-Saediet al. [49]. Here, cells are modeled as interacting Brownian particles. The derivation

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incorporates the possibility of different cell species and phenotypes. A central propertyof cells that is absent for other Brownian particles is that they can reproduce and die.Hence, an additional term for this nonconserved part of the dynamics has to be added.For describing tumor growth, the DDFT is coupled to a reaction-diffusion equation forthe density of a nutrient, whose availability determines the birth and death rates [46].By considering a system of two cell species, the competition of healthy and cancer cellscan also be studied [49]. A general overview over models for cancer growth is givenin Ref. [1000]. Similarly, the dynamics of cell colonies can also be studied via PFCmodels that can be derived from DDFT [1001].

Another biophysical application of DDFT is protein diffusion in cell membranes.The membrane is a mixture of various types of lipids, which rearrange if a chargedmacromolecule adsorbs. This dynamics can be modeled in DDFT [472]. DDFT alsoserves as a basis for lipid bilayer models [473] and plays a role in biological solvationdynamics [1002]. Moreover DDFT-like methods have been applied in the derivation ofa dynamical model for Mongolian gazelles [1003]. A DDFT model for disease spreading[50] is discussed in Section 8.2.6. Finally, DDFT is used to study ion channels [351,352](see Section 8.1.10 for details).

8.2. Phenomena

8.2.1. Relaxation

As discussed in Section 3.5.4, DDFT is a theory that describes the diffusive relaxationof colloidal systems towards an equilibrium state corresponding to the minimum of thefree energy. Hence, a natural application of DDFT is to model relaxation dynamics,i.e., the way in which a system moves to an equilibrium state after initially being outof equilibrium. For nonspherical particles, DDFT can be used to describe orientationalrelaxation, which was one of the main applications of early forms of DDFT [112,121,488] (see Section 4.6.1).

Relaxation dynamics was studied by Bier and van Roij [183] for platelike colloids.Here, a fluid is perturbed by a laser beam from an initial isotropic state. Depending onlaser power and initial chemical potential, different relaxation paths are observed. Thesame model is applied in Ref. [489] to a system connecting two reservoirs. Relaxingbinary fluids in a cavity were studied by Archer [272]. For certain systems, relaxationcan also have the form of a gravitational collapse [301,302,316,656]. Uematsu andYoshimori [490] analyzed the polarization relaxation of molecular liquids in externalfields. Bier and Arnold [1004] compared relaxation processes of interfaces and bulkphases. In Ref. [1005], nonequilibrium surface tension is considered. If the equilibriumstate is uniform, interfaces and thus surface tension are nonequilibrium phenomena.Their time evolution can be calculated in DDFT (see Ref. [1006] for another discussionof nonequilibrium surface tension). Relaxation of colloid-polymer mixtures was studiedby Stopper et al. [203]. Kruger and Dean [1007] used DDFT as a comparison to testthe relaxation dynamics of a theory of density fluctuations. Nunes et al. [306] analyzedthe relaxation of transient demixed states in a binary mixture. As a direction for futureresearch, the study of nonequilibrium fluctuations in DDFT was suggested by Giavazziet al. [1008].

Studying relaxation dynamics is also interesting for relating DDFT to glassy dynam-ics as described by MCT. In general, the van Hove function provides a good measurefor relaxation to equilibrium [31]. Its calculation is a typical application of DDFT(see Section 8.3). For glasses, a two-stage relaxation of the self-part of the van Hove

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function with a diverging relaxation time is characteristic. As shown in Ref. [342],this relaxation behavior can be recovered in DDFT. Berry and Grant [501] foundagreement between stochastic DDFT and MCT. Reinhardt and Brader [285] have,however, argued that DDFT is not capable of describing the nonergodic transition ofsimple MCT. (This point is discussed in Section 8.1.5.) Moreover, deviations from thetrue relaxation behavior can result from neglecting memory effects and superadiabaticforces [31,287,467]. The validity of DDFT in the description of relaxational dynamicswas explored by Penna and Tarazona [312] by comparisons to BD simulations.

8.2.2. Phase separation

Phase separation in colloidal fluids can occur through nucleation (if they are quenchednear the binodal) or spinodal decomposition (if they are quenched inside the binodal).The investigation of spinodal decomposition was important for the early stages in thedevelopment of DDFT and its predecessors [8,91,93,94]. In Ref. [14], spinodal decom-position is studied based on DDFT. A linear stability analysis (see Paragraph 7.2.1.1)can be used to model the early stages of phase separation. Linear stability analysiscan also reveal the existence of different ways for the homogeneous state to becomeunstable (spinodal and freezing modes) [311,727]. For the intermediate stages, a non-linear theory is required. A comparison between DDFT and experiments was made byZvyagolskaya et al. [353].

In Ref. [272], DDFT is applied to Gaussian core model (GCM) binary fluids ex-hibiting liquid-liquid phase separation. Both bulk phase separation and microphaseseparation occur. A symmetric GCM fluid is considered in Ref. [303]. If the differentparticle types have symmetric properties, the ensemble-averaged one-body profile isthe same for both of them if the confining potentials are identical. DDFT is able topredict the phase separation if the symmetry of the confining potentials is broken.In addition, DDFT can be applied to gas-liquid phase separation. This gives insightsinto the influence of phase separation on sedimentation [336]. Colloids with capillaryinteractions can be found to have a phase diagram that involves not only spinodal de-composition, but also a gravitation-like collapse [301,302]. Such a collapse was studiedfor a uniform disklike initial distribution by Bleibel et al. [316]. A detailed discussionof the gravitational collapse of Brownian particles can be found in Refs. [656,1009]. Forbinary fluids in a gravitational field, the interplay between thermodynamics and sedi-mentation can lead to a rich phase diagram [465]. In studies of 1D hard-sphere fluids,conditions for the coexistence of 1D and quasi-0D states can be found to be analo-gous to conditions for phase separation [1010]. Coarsening modes (Ostwald ripeningand translational modes) are studied in Ref. [727]. DDFT models including hydro-dynamics can also exhibit phase separation, as shown for a van der Waals fluid inRef. [402]. In dilute binary mixtures, phase separation can lead to the emergence ofbubbles that show damped oscillations [451]. Nonequilibrium phase separation undershear is discussed in Ref. [202]. For otherwise identical particles with different Stokescoefficients, external fields can lead to demixing [306]. DDFT is also applied to phaseseparation of particles on a sphere [454]. Finally, fluid demixing on spherical surfacescan be analyzed using Minkowski functionals [455].

Local phase transitions are considered in Refs. [422,432]. Behind a driven tracerparticle, a colloid-poor phase emerges that eventually forms a bubble [432]. Cavitation,which is a local phase separation, results from the pressure being higher in frontof a solid particle being driven through a fluid than behind it [422]. Microrheology,including DDFT applications, was reviewed in Ref. [423].

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Phase separation is also modeled using DDFT in more complex fluids. A mix-ture of particles with opposing adsorption preferences can show a colloidal-liquid-colloidal-liquid demixing transition driven by critical fluctuations [353]. Magnetic par-ticles, with have the magnetic moment as an additional degree of freedom, can alsoshow phase separation [448]. Spinodal decomposition in a magnetic fluid is consideredin Ref. [450]. In Ref. [726], spinodal decomposition is studied in a system with ananisotropic diffusion tensor. Microphase separation in polymer systems is discussedin Refs. [360,444,446,888]. Regarding nonequilibrium phase separation, DDFT is usedin the context of lane formation [323–325,332,431,498] (see Section 8.2.3). A studyof nonequilibrium phase separation in PFT can be found in Ref. [32] (see also Ref.[1011]). In Refs. [23,205], DDFT was applied to phase separation and phase equilibriain active systems. Phase coexistence of ABPs was moreover studied by Paliwal et al.[299] using a chemical-potential-like function. Actively switching colloids were studiedin Ref. [418].

Polymer DDFT (see Section 3.2.3) is a very frequently used method in the descrip-tion of phase separation in polymer systems, in particular copolymer melts [55,360–362,433,524,830–865].

When studying phase separation in DDFT, the problem of averaging explained inSection 3.5.2 is of potential importance. This was discussed in detail by Bleibel et al.[316] who studied the capillary collapse of a uniform disk of particles and comparedthe time evolutions predicted by DDFT with and without an average over the initialdistribution of the particles. The results do not agree, since the random initial distri-bution of particles that is captured only by a density profile not averaged over initialconditions can lead to the transient formation of dense regions. This aspect can berelevant for spinodal decomposition, since even for an unstable bulk state an averageover initial conditions may lead to a constant density profile and then to ∂tρ = 0. Thetypical solution for this problem, which is to add a small perturbation to the constantdensity profile (see Paragraph 7.2.1.1), does not always give accurate results [316].

8.2.3. Pattern formation

Pattern formation, which can be a consequence of phase separation, is one of thecentral effects studied in soft matter theory. As a dynamical theory for the spatialdistribution of the particles, DDFT is also well suited to describe pattern formation.A good example is lane formation, which is a nonequilibrium phase transition [1012].Laning can arise, e.g., in colloidal mixtures where different particle types are drivenin different ways by an external force [324]. It can be analyzed using linear stabil-ity analysis [324,325,431]. Moreover, lane formation can occur on short time scales insystems that are subject to time-dependent shear [429]. Laning can also be observedin active matter [498]. Superadiabatic effects in lane formation are considered in Ref.[332]. DDFT has also been used to describe reentrance effects [325] and fluids with at-tractive interactions [323]. A particularly frequently used method for studying patternformation is polymer DDFT [56,269,270,866–881].

Pattern formation can also occur in evaporation/dewetting processes in thin films(see also Section 8.1.4). Here, effects such as fingering can occur as a consequence of theinterplay between transport and evaporation processes. Suspensions of nanoparticles influids are modeled by a DDFT for mixtures that also contains nonconserved terms forevaporation. Tools employed here are both linear stability analysis and the numericalsolution of the full nonlinear DDFT [457–459].

Recently, DDFT has been applied to quasicrystal formation in soft matter [732,733].

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Figure 8. Time-evolving density field ρ(~r, t) with mean value ρ0 in a DDFT model for solidification. Animposed nucleation cluster grows to a crystal. Adapted from Fig. 2 from Ref. [1023].

Self-organization into quasicrystals requires the presence of two different dominantwavenumbers (although other wavenumbers can be important too [732]). DDFT al-lows, by means of a linear stability analysis, to identify important wavenumbers, giv-ing insights into which length scales are important. Archer et al. [730, 731] foundquasicrystal formation in DDFT to be the result of linear growth of one wavenumberfollowed by the nonlinear selection of another one. A strategy for finding quasicrys-tals was proposed in Ref. [1013]. Quasicrystals have also been studied in PFC models[1014–1016].

In colloidal systems, DDFT is used to study, e.g., the design of spatiotemporal pat-terns by means of time-dependent external potentials [957] or traveling band forma-tion [345]. Stripe formation in magnetic fluids can occur as a consequence of intrinsicphase separation combined with external drives [449]. In a DDFT model for epidemicspreading (see Section 8.2.6), the separation of healthy and infected persons can beobserved [50]. Further applications of DDFT to pattern formation include patternsbehind fronts [676], crystallization on structured surfaces [196], Brownian particleswith repulsive soft-core interactions [244], band formation [309], activity-driven spins[461], and defect morphologies in polyethylene lattices [1017]. Images of pattern for-mation allow to extract constitutive relations [1018]. The investigation of localizedstates [1019–1021] is a possible direction for future work. A general review of patternformation is given by Ref. [1022].

8.2.4. Nucleation and solidification

A topic that is related to both relaxation and phase separation is the study of nucle-ation and solidification. Static DFT is a central tool for the study of nucleation andcrystallization [1024] (see Ref. [1025] for a review). It can be used as a basis for calcu-lating dynamical nucleation rates [371,1025,1026]. DDFT has also been applied to thisproblem. A brief review can be found in Ref. [195]. In nucleation theory, the criticalnucleus can be determined from DFT and is then used as a starting point for a timeevolution in DDFT [45]. The initial nucleation cluster can also be imposed, DDFTthen predicts whether or not a crystal will grow [153]. DDFT is a useful tool for mod-eling heterogeneous nucleation [1027]. Effects that can be analyzed in DDFT includethe difference between rapid and slow heating [1027], freezing/melting in interfacialsystems [901], crystallization with multiple seeds and grain boundary formation [197],nucleation and growth of ice crystals [207], formation of protein clusters [416,417],influence of confinement obstacles on nucleation [187], and self-assembly of nanopar-ticles [189]. Moreover, crystallization on patterned surfaces has been studied, wherethe interplay between crystal and substrate structures leads to “compatibility waves”[196]. Stochastic methods are also applied in this field [1028].

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An example is shown in Fig. 8 on the previous page, which is adapted from Ref.[1023]. Here, crystal growth at externally imposed nucleation clusters was studied.Figure 8 shows snapshots of the time evolution arising from a weakly compressedrhombic nucleation seed consisting of 19 tagged particles. As can be seen, DDFTdescribes the evolution to the crystalline equilibrium state.

Note that, however, many studies of nucleation processes are performed using PFCmodels rather than DDFT (see Refs. [364,365,674,1029–1034] for examples for andRefs. [1032,1035–1039] for an overview over the use of PFC models in nucleation the-ory). Although they involve more approximations than DDFT (and are thus quantita-tively less accurate [407]), PFC models are very successful in this area. Comparisonsof DDFT and PFC models in solidification can be found in Refs. [44,45,407,677]. Typ-ically, the agreement between PFC models and DDFT is better at longer wavelengths[1040]. Archer et al. [370] discussed in detail the effect of the PFC approximation onthe phase diagram (see Section 6.3). An overview over various theoretical and experi-mental approaches to nucleation is given by Ref. [54].

For solidification, DDFT allows, in contrast to DFT, to find not only the equi-librium profile, but also effects of the dynamical evolution on the structure. For ex-ample, DDFT has been applied to solidification fronts (see Paragraph 7.2.1.2). Thewavelengths selected by front propagation can differ from that of equilibrium crys-tals, leading to disorder [462,676,725]. This is particularly interesting for analyzingwhy some materials form glasses [462]. Moreover, phases that are developed duringfreezing, but which are not the equilibrium state, such as quasicrystals, can also beobserved in DDFT [730,731]. The freezing transition can also be analyzed in kineticextensions of DDFT [502]. Hou et al. [407] considered self-healing effects in crystalgrowth, where, despite initial impurities being present, particles rearrange to the idealcrystal structure. Finally, even if one is only interested in the long-time equilibriumstate, DDFT can be used as a minimization procedure and thus as a tool for DFT[1041].

In general, nucleation involves that an energy barrier is crossed which has its maxi-mum at the critical nucleus size. A method for the calculation of nucleation pathwaysis to find the “minimal free energy path”, with is the most likely route for nucleation[1042,1043]. A particularly interesting strategy was suggested by Lutsko [258,1044–1046] for obtaining the “most likely path” for nucleation. The basic idea originatesfrom Onsager and Machlup [1047]. DDFT, being a theory in which the free energymonotonously decreases, can (without noise terms) not describe energy barrier cross-ing [1048]. One can, however, start at the top of the barrier and then perform a DDFTtime evolution forwards in time to the final state and backwards in time to the initialstate (both going downwards the energy barrier), thereby determining the most likelynucleation pathway.

8.2.5. Chemical reactions

Reaction-diffusion equations, which are obtained by adding reaction terms to the dif-fusion equation (37), have a long tradition in the description of effects such as patternformation [1049]. Similar reaction models can also be obtained using the Cahn-Hilliardequation (218) rather than the diffusion equation [1050]. Since DDFT is a generalizeddiffusion equation, one can incorporate chemical reactions by extending DDFT to areaction-diffusion equation by adding reaction terms [416,417]. This is the idea behind“reaction-diffusion DFT” (RDDFT), which is applied to NO oxidation in Ref. [52].Reactions can only occur if the reacting particles meet, such that simple models are

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constructed through reaction-diffusion equations in which the number of encounters islimited by diffusion but not by interactions. However, in particular in crowded envi-ronments such as cells, the interactions of the particles have effects on the frequency ofencounters [1051]. This problem can be studied in DDFT, which allows to incorporatesuch effects [822]. Similarly, DDFT can be used to analyze the effects of the productspecies on diffusion-controlled reactions. Products can, e.g., accumulate at catalystsand thereby interact with the reactants which might have very different physical prop-erties [452]. Diffusion-controlled reactions are moreover discussed in Ref. [307] (basedon a linear theory related to DDFT). Besides these general studies, the formationand stability of protein clusters has been studied using DDFT [417]. A more accuratemodel takes into account, also within DDFT, that the reaction rates are not fixed butstate dependent [416]. Recently, RDDFT was applied to study active switching, theswitches then correspond to the reaction terms [418]. It also plays a role in the descrip-tion of surfactant-covered interfaces [419]. Nanoreactors [1052] and crowded systemsunder shear [1053] are an additional possible field of application. Nonideal diffusionalso plays a role in Brownian aggregation, where FMT and DDFT can be used topredict aggregation rate constants [190,191].

8.2.6. Disease spreading

Closely related to RDDFT is the DDFT model for disease spreading proposed by teVrugt et al. [50]. It is an extension of the simple SIR model [1054] used in theoreticalbiology, which describes the temporal evolution of the number of susceptible persons S,infected persons I, and recovered persons R under the assumptions that susceptiblepersons are infected at a rate cir when meeting infected persons and that infectedpersons recover at a rate crr and are immune afterwards. In an extension known asSIRD model, it is additionally assumed that infected persons die at a rate cdr. TheSIR-DDFT model is given by

∂

∂tS(~r, t) = ΓS ~∇ ·

(S(~r, t)~∇ δF

δS(~r, t)

)− cirS(~r, t)I(~r, t), (273)

∂

∂tI(~r, t) = ΓI ~∇ ·

(I(~r, t)~∇ δF

δI(~r, t)

)+ cirS(~r, t)I(~r, t)− crrI(~r, t)− cdrI(~r, t), (274)

∂

∂tR(~r, t) = ΓR~∇ ·

(R(~r, t)~∇ δF

δR(~r, t)

)+ crrI(~r, t) (275)

and describes persons as interacting particles, where repulsive interactions containedin the functional F incorporate effects of social distancing and quarantine. The mobili-ties ΓS , ΓI , and ΓR of susceptible, infected, and recovered persons, respectively, can bedifferent. Numerically, it can be shown that at certain interaction strengths the modelundergoes a transition to a phase in which the number of infected persons is signif-icantly reduced as a consequence of isolation and social distancing. The interactionslead to a pattern formation effect in which infected and healthy persons separate. Anextension of the SIR-DDFT model incorporating adaptive political interventions wasdeveloped in Ref. [420], where the effects of various containment strategies on diseasespreading and the conditions under which multiple waves of a pandemic occur areinvestigated.

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8.2.7. Feedback control

Another possible reason for pattern formation is time-delayed feedback, as shown byTarama et al. [345] for traveling bands. DDFT is a useful tool for modeling effects oftime-delayed feedback in systems with interactions. In time-delayed feedback control,a (nonlinear) system is controlled using forces that depend on the state of the systemat a previous time t− τd where τd is a delay. A current topic of research is the way inwhich such control mechanisms are affected by particle interactions in the controlledsystem. This topic is reviewed in Ref. [1055]. DDFT allows to incorporate particleinteractions [809]. As shown by Lichtner and Klapp [1056], a current reversal can beobserved in interacting colloidal systems controlled by time-delayed feedback. Gernertand Klapp [1057] proposed a feedback control scheme for enhancing colloidal motilityin interacting systems. Grawitter and Stark [419] explored a feedback mechanism cou-pling flow velocities to external light spots in fluids with photoresponsive surfactants(see also Ref. [1058]).

8.2.8. Sound waves

In Ref. [131], the speed of sound in an atomic fluid is calculated using the DDFT foratomic fluids (see Section 4.7.1). Taylor expanding the excess free energy, decomposingthe density field as ρ(~r, t) = ρ0 + ∆ρ(~r, t), and using Eq. (148) gives

∂2

∂t2∆ρ(~r, t) + ν

∂

∂t∆ρ(~r, t) =

kBT

m~∇2∆ρ(~r, t)− kBTρ0

m~∇2

∫d3r′∆ρ(~r′, t)c(2)(~r − ~r′).

(276)Making the plane-wave ansatz

∆ρ(~r, t) = eei(~k·~r−ωt) (277)

with a small amplitude e allows to obtain the dispersion relation for the consideredfluid. Setting ~k ·~k = q2 + 2iq/` with q, ` ∈ R (which corresponds to assuming `−1 q,i.e., to assuming that propagation and attenuation length scales are well separated)and separating real and imaginary parts in Eq. (276) gives the dispersion relation

ω2(q) =q2kBT

mS(q)(278)

and the attenuation length

`(q) =2ω

νq. (279)

In the long-wavelength limit, we obtain the speed of sound

cs =ω

q=

1√ρ0kBTχT

(280)

with the isothermal compressibility χT , when using that S(0) ≈ ρ0kBTχT [149]. Dueto the negligence of temperature fluctuations in Eq. (148), the result (280) is only validat low temperatures and high densities. In Ref. [428], sound propagation in confinedfluids perpendicular to walls is discussed using a DDFT with momentum density.

111

While these approaches consider fluids, DFT is also useful for the description ofwaves in crystals. It allows to calculate the free energy change associated with asmall deformation of a crystal, which then gives the phonon dispersion relation [1059–1062]. More recently, a combination of DFT and projection operator methods (seeSection 5.3.1) was employed in Refs. [537,538]. The relevant variables include, in astate with spontaneously broken symmetry, not only the conserved hydrodynamicvariables, but also symmetry-restoring variables. For the theory developed in Refs.[537,538], the relevant variables are therefore given by the density fluctuations nearthe reciprocal lattice vectors and by the momentum density. If one considers the lin-ear response regime and neglects dissipation, the projection operator formalism giveslinear reversible equations of motion where the transport coefficients are given byequilibrium correlation functions. These can be evaluated using DFT.

8.3. Dynamics of the van Hove function

Finally, we present a “theoretical application” of DDFT, namely the investigation ofthe van Hove function. This is connected to aspects discussed in previous sections,since the van Hove function is important, e.g., for relaxation dynamics (Section 8.2.1)and electrochemistry (Section 8.1.10). Two further theoretical applications of DDFT,namely the derivation of MCT and PFC models, are discussed in Sections 6.1 and 6.3,respectively.

The van Hove function [1063] is a quantity used in the description of diffusionprocesses. It can be measured through confocal microscopy and scattering techniques[203]. For a system of N particles, the van Hove function is given by

GvH(~r, t) =1

N

⟨ N∑i=1

N∑j=1

δ(~r + ~rj(0)− ~ri(t))⟩

(281)

and measures the probability of finding an arbitrary particle i located at position ~rat time t given that a particle j has been located at the origin at t = 0 [194]. Thevan Hove function can be splitted into a self-part and a distinct part, distinguishingbetween the cases i = j and i 6= j [159].

DDFT methods were applied to calculate van Hove functions in ion systems in Refs.[117–120,406,641,963]. Here, a generalized transport equation including momentumand memory (similar to Eqs. (62) and (63)), which is obtained from DDFT, is used. Theunknown memory kernel is assumed to be equal to the ion friction (see Section 8.1.10).On this basis, a closed equation for the van Hove function of an electrolyte system canbe obtained, which is most conveniently dealt with in Fourier-Laplace space [117].

A method for studying the dynamics of the van Hove function in DDFT via ageneralization of Percus’ test particle limit [74] to dynamical correlation functions wasderived by Archer et al. [343]. The idea is that a one-component system can be thoughtof as a binary mixture of two species s and d with identical interactions and properties.The species s only consists of one particle. This method allows to keep track of acertain “test particle”. The self-part and the distinct part of the van Hove function canthen be identified with the one-body densities. Their dynamics is described by DDFT[343]. This method has been applied to complex fluids [358], systems of colloidal hardspheres [159,293,342], one-dimensional hard rods [285], colloid-polymer mixtures [203],Brownian disks [204], and polymer nanocomposites [51]. A comparison to experimentsis made in Ref. [204]. An extended dynamical theory for the van Hove function has

112

been obtained by using PFT instead of DDFT [27]. The van Hove function can alsobe determined from the nonequilibrium Ornstein-Zernike relation derived by Braderand Schmidt [24]. Superadiabatic van Hove currents are calculated in Ref. [25].

9. Outlook

After having reviewed the existing literature on DDFT, we here give an outlook overimportant directions of future work.

An interesting idea would be the application of machine learning methods to DDFT,as it is already successfully done in quantum DFT. Machine learning methods can beused to find approximate free energy functionals [794] that are required as an inputfor DDFT.

On top of that, the various lines of DDFT presented in Section 3 have had a relativelyindependent development. In particular, articles using or extending the DDFT byMarini Bettolo Marconi and Tarazona [12] rarely discuss work based on the DDFTby Fraaije [10] and vice versa, even though both methods are very widely used. Thisis a missed opportunity, since results obtained in one “strain” of DDFT can also beuseful for the other one. The method by Marini Bettolo Marconi and Tarazona [12]has undergone a rich theoretical development, while simulation methods based on thework by Fraaije et al. [124] are now used in polymer industry. Combinations wouldoffer further room for both theory and applications.

Moreover, applications of DDFT have, up to now, been focused on various types ofliquids. While this is not surprising given that this is what the theory is constructedfor, applications to quantum and astrophysical systems [28,47,328] show that the gen-eral principle can also be applied in other contexts. Further exploring the (possible)connections to theories for many-particle quantum or astrophysical systems could thusinspire a huge number of extensions as well as applications of classical DDFT. On theother hand, methods that are established in quantum mechanics, such as the Car-Parrinello approach [1064], could have potential for classical DDFT [1065].

In soft matter physics, applications in biological contexts have a lot of unexploredpotential. DDFT, as a microscopic theory of complex fluids that is applicable to non-spherical particles, confinement, and crowded environments, is well suited to describeintracellular diffusion as well as transport with blood flow. On larger scales, DDFTmodels for active matter have, of course, a natural application for studying biologicalmicroswimmers and active liquid crystals. Applications of classical DFT to crowds[1066] as well as descriptions of population dynamics by dynamic equations for theone-body density [1003] also indicate the dynamics of crowds or flocks as a possibleapplication of active DDFTs.

Moreover, applications of DDFT to driven and active systems tend to face theproblem that DDFT is based on an adiabatic approximation which is not generallypossible for such systems. This requires the construction and development of moregeneral methods (such as PFT) as well as the extension of DDFT. A combination ofanalytical and numerical work can be applied here. For example, in active systemspair-correlation functions [1067,1068] obtained from simulation results can be usedas an input for active DDFTs to replace the equilibrium approximation. Similarly,numerical simulations allow to obtain flow fields [1069] required for hydrodynamictheories.

Furthermore, given that DDFT is a paradigmatic example of a theory describ-ing the collective dynamics of nonequilibrium systems, studying the foundations of

113

DDFT provides interesting insights into general problems of nonequilibrium statisti-cal mechanics. One such problem is the importance of memory effects. As discussed inSection 5, these are naturally incorporated in theories derived using power functionaltheory or the projection operator formalism. Standard deterministic DDFT arises asa limiting case after certain simplifications that include neglecting memory effects.Additional topics for which basic research on DDFT can be expected to be fruitful arethe description of nonergodic systems [39] (see Section 8.1.5), the approach to thermo-dynamic equilibrium [60] (see Section 3.5.4), and origin and importance of fluctuations[211] (see Section 3.5.2). A field in which all these aspects are relevant is the study ofthe glass transition (see Section 8.1.5), which can greatly benefit from an improved un-derstanding of the complex relationship between microscopic dynamics, deterministicDDFT, stochastic DDFT, and MCT.

Finally, a lot of unexplored potential can be found in the applications of gener-alizations of DDFT. Although very general methods have been developed, allowingto include, e.g., nonspherical particles or nonisothermal systems, most applications ofDDFT use its simple standard form. Considering more general forms would allow formuch more general insights, e.g., into the relaxation and transport dynamics of sys-tems consisting of anisotropic particles, which are a very promising field of research insoft matter physics. This also opens further room for industrial applications of DDFT[1070].

10. Conclusions

We have presented a thorough review of classical DDFT, including its relation to othertheories and methods as well as its extensions and applications. DDFT is, as shouldbe clear from our presentation, a useful and versatile method as well as an active fieldof research. It will continue to shape work in statistical mechanics and soft matterphysics in the years to come.

Acknowledgments

We thank Jens Bickmann, Johannes G. E. M. Fraaije, Kazuhiro Fuchizaki, Ben God-dard, Matthieu Marechal, Roland Roth, Kenneth S. Schweizer, Peter Stratmann, UweThiele, and Rene Wittmann for helpful discussions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Funding

H.L. and R.W. are funded by the Deutsche Forschungsgemeinschaft (DFG, GermanResearch Foundation) – LO 418/25-1; WI 4170/3-1.

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